A NONLOCAL p−LAPLACIAN EVOLUTION EQUATION WITH NON HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

F. ANDREU, J. M. MAZÓN, J. D. ROSSI AND J. TOLEDO

Abstract. In this paper we study the nonlocal p−Laplacian type diffusion equation

u_{t}(t, x) = ∫_{ℝ^{N}} J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy, (t, x) ∈]0, T [×Ω,

with u(t, x) = ψ(x) for (t, x) ∈]0, T [×(ℝ^{N} \ Ω). If p > 1, this is the nonlocal analogous problem to the well known local p−Laplacian evolution equation u_{t} = div(|∇u|^{p−2}∇u) with Dirichlet boundary condition u(t, x) = ψ(x) on (t, x) ∈]0, T [×∂Ω. If p = 1, this is the nonlocal analogous to the total variation flow. When p = +∞ (this has to be interpreted as the limit as p → +∞ in the previous model) we find an evolution problem that can be seen as a nonlocal model for the formation of sandpiles (here u(t, x) stands for the height of the sandpile) with prescribed height of sand outside of Ω.

We prove, as main results, existence, uniqueness, a contraction property that gives well posedness of the problem and the convergence of the solutions to solutions of the local analogous problem when a rescaling parameter goes to zero.

April 17, 2008

1. Introduction

In this paper we study the nonlocal diffusion equation

u_{t}(t, x) = ∫_{ℝ^{N}} J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy (t, x) ∈]0, T [×Ω,

where Ω is a bounded domain and u is prescribed in ℝ^{N} \ Ω as u(t, x) = ψ(x) for (t, x) ∈ ]0, T [×(ℝ^{N} \ Ω). We consider 1 < p < +∞ as well as the extreme cases p = 1 and the limit p ↗ +∞. Throughout the paper, we assume that J : ℝ^{N} → ℝ is a nonnegative, radial, continuous function, strictly positive in B(0, 1), vanishing in ℝ^{N} \ B(0, 1) and such that ∫_{ℝ^{N}} J(z) dz = 1.

First, let us briefly introduce the prototype of nonlocal problem that will be considered along this work. Nonlocal evolution equations of the form

(1.1)  u_{t}(t, x) = (J ∗ u − u)(t, x) = ∫_{ℝ^{N}} J(x − y)u(t, y) dy − u(t, x),

and variations of it, have been recently widely used to model diffusion processes. More precisely, as stated in [32], if u(t, x) is thought of as a density at the point x at time t and J(x − y) is thought of as the probability distribution of jumping from location y to location x, then ∫_{ℝ^{N}} J(y − x)u(t, y) dy = (J ∗ u)(t, x) is the rate at which individuals are arriving at position x from all other places and −u(t, x) = − ∫_{ℝ^{N}} J(y − x)u(t, x) dy is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation (1.1). For recent references on nonlocal diffusion see, [6], [7], [8], [11], [13], [14], [21], [22], [23], [24], [25], [26], [27], [28], [32], [34], [37] and references therein.

Key words and phrases. Nonlocal diffusion, p−Laplacian, Non homogeneous Dirichlet boundary conditions, Total variation flow, Sandpiles.

2000 Mathematics Subject Classification. 35B40, 45A07, 45G10.

The first goal of this paper is to study the following nonlocal nonlinear diffusion problem,

P_{p}^{J}(u_{0}, ψ) \{ \begin{matrix}u_{t}(t, x) = ∫_{Ω} J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy \\ + ∫_{Ω_{J} \\overline{Ω}} J(x − y)|ψ(y) − u(t, x)|^{p−2}(ψ(y) − u(t, x)) dy, (t, x) ∈]0, T [×Ω, \\ u(0, x) = u_{0}(x), x ∈ Ω. \end{matrix}

Here Ω_{J} = Ω + supp(J) and ψ is a given function ψ : Ω_{J} \ \overline{Ω} → ℝ.

Observe that we can rewrite P_{p}^{J}(u_{0}, ψ), setting u(t, x) = ψ(x) in Ω_{J} \ \overline{Ω}, as

\begin{cases}u_{t}(t, x)=∫_{Ω_{J}}J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy, (t, x) ∈]0, T [×Ω, \\ u(t, x) = ψ(x), (t, x) ∈]0, T [×(Ω_{J} \ \overline{Ω}), \\ u(0, x) = u_{0}(x), x ∈ Ω, \end{cases}

and we call it the nonlocal p-Laplacian problem with Dirichlet boundary condition. Note that we are prescribing the values of u outside the domain Ω and not only on its boundary. This is due to the nonlocal character of the problem.

Let us state the precise definition of solution. Solutions to P_{p}^{J}(u_{0}, ψ) will be understood in the following sense.

Definition 1.1. Let 1 < p < +∞. A solution of P_{p}^{J}(u_{0}, ψ) in [0, T] is a function

u ∈ C([0, T]; L^{1}(Ω)) ∩ W^{1,1}(]0, T [; L^{1}(Ω))

which satisfies u(0, x) = u_{0}(x) a.e. x ∈ Ω and

\begin{aligned}u_{t}(t, x) & = ∫_{Ω} J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy \\ & + ∫_{Ω_{J} \\overline{Ω}} J(x − y)|ψ(y) − u(t, x)|^{p−2}(ψ(y) − u(t, x)) dy, \end{aligned}

for a.e t ∈]0, T [ and a.e. x ∈ Ω.

Our first result shows existence and uniqueness of a global solution for this problem. Moreover, a contraction principle holds.

Theorem 1.2. Assume p > 1 and let u_{0} ∈ L^{p}(Ω), ψ ∈ L^{p}(Ω_{J} \ \overline{Ω}). Then, there exists a unique solution to P_{p}^{J}(u_{0}, ψ) in the sense of Definition 1.1. Moreover, if u_{i}_{0} ∈ L^{1}(Ω) and u_{i} is a solution in [0, T] of P_{p}^{J}(u_{i}_{0}, ψ), i = 1, 2, respectively. Then

∫_{Ω}(u_{1}(t) − u_{2}(t))^{+} ≤ ∫_{Ω}(u_{1}_{0} − u_{2}_{0})^{+} for every t ∈ [0, T].

If u_{i}_{0} ∈ L^{p}(Ω), i = 1, 2, then

∥u_{1}(t) − u_{2}(t)∥_{L^{p}(Ω)} ≤ ∥u_{1}_{0} − u_{2}_{0}∥_{L^{p}(Ω)} for every t ∈ [0, T].

A NONLOCAL p−LAPLACIAN EVOLUTION EQUATION 3

Our next step is to rescale the kernel J appropriately and take the limit as the scaling parameter goes to zero. To be more precise, for p > 1, we consider the local p−Laplace evolution equation with Dirichlet boundary condition

D_{p}(u_{0}, ψ̃) \{ \begin{matrix}u_{t} = Δ_{p}u & in ]0, T [×Ω, \\ u = ψ̃ & on ]0, T [×∂Ω, \\ u(0, x) = u_{0}(x) & in Ω, \end{matrix}

where the boundary datum ψ̃ is assumed to be the trace of a function defined in a larger domain and the operator in the equation, Δ_{p}u = div(|∇u|^{p−2}∇u), is the usual local p-Laplacian.

We prove that the solutions of this local problem can be approximated by solutions of a sequence of nonlocal p-Laplacian problems of the form P_{p}^{J}. Indeed, for given p ≥ 1 and J we consider the rescaled kernels

(1.2)  J_{p,ε}(x) := \frac{C_{J,p}}{ε^{p+N}} J (\frac{x}{ε}) ,  where  C_{J,p}^{−1} := \frac{1}{2} ∫_{ℝ^{N}} J(z)|z_{N}|^{p} dz

is a normalizing constant in order to obtain the p-Laplacian in the limit instead a multiple of it, and we obtain the following result.

Theorem 1.3. Let Ω be a smooth bounded domain in ℝ^{N} and ψ̃ ∈ W^{1/p′,p}(∂Ω) ∩ L^{∞}(∂Ω). Let ψ ∈ W^{1,p}(Ω_{J})∩L^{∞}(Ω_{J}) such that ψ|_{∂Ω} = ψ̃. Assume J(x) ≥ J(y) if |x| ≤ |y|. Let T > 0 and u_{0} ∈ L^{p}(Ω). Let u_{ε} the unique solution of P_{p}^{J_{p,ε}}(u_{0}, ψ) and u the unique mild-solution of D_{p}(u_{0}, ψ̃). Then

(1.3)  lim_{ε→0} sup_{t∈[0,T]} ∥u_{ε}(t, .) − u(t, .)∥_{L^{p}(Ω)} = 0.

Note that the above result says that P_{p}^{J} is a nonlocal problem analogous to the p−Laplacian with Dirichlet boundary condition.

The second goal of this paper is to study the Dirichlet problem for p = 1, called the nonlocal total variation flow, which can be written formally as

P_{1}^{J}(u_{0}, ψ) \{ \begin{matrix}u_{t}(t, x) & = ∫_{Ω} J(x − y)\frac{u(t, y) − u(t, x)}{|u(t, y) − u(t, x)|} dy \\ & + ∫_{Ω_{J} \Ω} J(x − y)\frac{ψ(t, y) − u(t, x)}{|ψ(t, y) − u(t, x)|} dy, (t, x) ∈]0, T [×Ω, \\ & u(0, x) = u_{0}(x), x ∈ Ω. \end{matrix}

We give the following definition of what we understand by a solution of P_{1}^{J}(u_{0}, ψ).

Definition 1.4. A solution of P_{1}^{J}(u_{0}, ψ) in [0, T] is a function

u ∈ C([0, T]; L^{1}(Ω)) ∩ W^{1,1}(]0, T [; L^{1}(Ω))

which satisfies u(0, x) = u_{0}(x) a.e. x ∈ Ω and

u_{t}(t, x) = ∫_{Ω_{J}} J(x − y)g(t, x, y) dy a.e. in ]0, T [×Ω,

for some g ∈ L^{∞}(0, T; L^{∞}(Ω_{J} × Ω)) with ∥g∥_{∞} ≤ 1 such that for almost every t ∈]0, T [, g(t, x, y) = −g(t, y, x) and

J(x − y)g(t, x, y) ∈ J(x − y)sign(u(t, y) − u(t, x)), (x, y) ∈ Ω × Ω,
J(x − y)g(t, x, y) ∈ J(x − y)sign(ψ(y) − u(t, x)), (x, y) ∈ Ω × Ω_{J}.

To get existence and uniqueness of this kind of solutions, the idea is to take the limit as p ↘ 1 of solutions to P_{p}^{J} with p > 1.

Theorem 1.5. Let u_{0} ∈ L^{1}(Ω) and ψ ∈ L^{1}(Ω_{J} \ \overline{Ω}). Then, there exists a unique solution to P_{1}^{J}(u_{0}) in the sense of Definition 1.4. Moreover, if u_{i}_{0} ∈ L^{1}(Ω) and u_{i} are a solutions in [0, T] of P_{1}^{J}(u_{i}_{0}), i = 1, 2. Then

∫_{Ω}(u_{1}(t) − u_{2}(t))^{+} ≤ ∫_{Ω}(u_{1}_{0} − u_{2}_{0})^{+} for every t ∈ [0, T].

In this case we can rescale the kernel as in (1.2) in order to obtain convergence of the solutions of the corresponding rescaled problem to the solution of the Dirichlet problem for the total variational flow, that is,

D_{1}(u_{0}, ψ̃) \{ \begin{matrix}u_{t} = div (\frac{Du}{|Du|}) & in ]0, T [×Ω, \\ u = ψ̃ & on ]0, T [×∂Ω, \\ u(0, x) = u_{0}(x) & in Ω. \end{matrix}

Theorem 1.6. Let Ω be a smooth bounded domain in ℝ^{N}. Assume J(x) ≥ J(y) if |x| ≤ |y|. Let T > 0, u_{0} ∈ L^{1}(Ω), ψ̃ ∈ L^{∞}(∂Ω) and ψ ∈ W^{1,1}(Ω_{J} \\overline{Ω})∩L^{∞}(Ω_{J} \\overline{Ω}) such that ψ|_{∂Ω} = ψ̃. Let u_{ε} be the unique solution of P_{1}^{J_{1,ε}}(u_{0}, ψ). Then, if u is the unique solution of D_{1}(u_{0}, ψ̃),

lim_{ε→0} sup_{t∈[0,T]} ∥u_{ε}(t, .) − u(t, .)∥_{L^{1}(Ω)} = 0.

Finally, the third goal of this paper is to study the limit case p = +∞, that has to be understood as the limit of our nonlocal evolution problems as p → +∞ (see Section 4). In this case we recover a nonlocal model for the evolution of sandpiles which is the nonlocal version of the Prigozhin model [36]. Then, the nonlocal limit problem with source for p = +∞ can be written as

P_{∞}^{J}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{t}(t, ·) ∈ ∂G_{∞,ψ}^{J}(u(t)), & a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x), \end{matrix}

where G_{∞,ψ}^{J} is the functional

G_{∞,ψ}^{J}(u) = \{ \begin{matrix}0 & if |u(x) − u(y)| ≤ 1, for x, y ∈ Ω \\ & and |ψ(y) − u(x)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ & with x − y ∈ supp(J) \\ +∞ & in other case, \end{matrix}

that is, G_{∞,ψ}^{J} = I_{K_{∞,ψ}^{J}} the indicator function of the set

K_{∞,ψ}^{J} := \{u ∈ L^{2}(Ω) : \begin{matrix}|u(x) − u(y)| ≤ 1, x, y ∈ Ω \\ and |ψ(y) − u(x)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ with x − y ∈ supp(J) \end{matrix} \} .

More precisely, we obtain the following result.

Theorem 1.7. Let ψ ∈ L^{∞}(Ω_{J} \ \overline{Ω}) such that K_{∞,ψ}^{J} ≠ ∅. Let T > 0, f ∈ L^{2}(0, T; L^{2}(Ω)), u_{0} ∈ L^{2}(Ω) ∩ L^{p}(Ω) such that u_{0} ∈ K_{∞,ψ}^{J}, and u_{p} the unique solution of the nonlocal p-Laplacian with a source term f, P_{p}^{J}(u_{0}, ψ, f) (see Section 4). Then, if u_{∞} is the unique solution to P_{∞}^{J}(u_{0}, ψ, f),

lim_{p→∞} sup_{t∈[0,T]} ∥u_{p}(t, ·) − u_{∞}(t, ·)∥_{L^{2}(Ω)} = 0.

Our next step is to rescale the kernel J appropriately and take the limit as the scaling parameter goes to zero. We will suppose that Ω is convex and ψ verifies ∥∇ψ∥_{∞} ≤ 1. For ε > 0, we rescale the functional G_{∞,ψ}^{J} as follows

G_{∞,ψ}^{ε}(u) = \{ \begin{matrix}0 & if |u(x) − u(y)| ≤ ε, for x, y ∈ Ω \\ & and |ψ(y) − u(x)| ≤ ε, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ & with |x − y| ≤ ε \\ +∞ & in other case, \end{matrix}

that is, G_{∞,ψ}^{ε} = I_{K_{∞,ψ}^{ε}}, where

K_{∞,ψ}^{ε} := \{u ∈ L^{2}(Ω) : \begin{matrix}|u(x) − u(y)| ≤ ε, x, y ∈ Ω \\ and |ψ(y) − u(x)| ≤ ε, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ with |x − y| ≤ ε \end{matrix} \} .

Consider the gradient flow associated to the functional G_{∞,ψ}^{ε}

P_{∞}^{ε}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{t}(t, ·) ∈ ∂I_{K_{∞,ψ}^{ε}}(u(t)), & a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x), & in Ω, \end{matrix}

and the limit problem

P_{∞}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{∞,t} ∈ ∂I_{K_{ψ}}(u_{∞}), & a.e. t ∈]0, T [, \\ u_{∞}(0, x) = u_{0}(x), & in Ω, \end{matrix}

where

K_{ψ} := {u ∈ W^{1,∞}(Ω) : ∥∇u∥_{∞} ≤ 1, u|_{∂Ω} = ψ|_{∂Ω}} .

Now we state our result concerning the limit as ε → 0 for the sandpile model (p = +∞).

Theorem 1.8. Assume Ω is a convex bounded domain in ℝ^{N}. Let T > 0, f ∈ L^{2}(0, T; L^{2}(Ω)), ψ ∈ W^{1,∞}(Ω_{J} \ \overline{Ω}) such that ∥∇ψ∥_{∞} ≤ 1, u_{0} ∈ W^{1,∞}(Ω) such that ∥∇u_{0}∥_{∞} ≤ 1 and u_{0}|_{∂Ω} = ψ|_{∂Ω} (this means u_{0} ∈ K_{ψ}), and consider u_{∞,ε} the unique solution of P_{∞}^{ε}(u_{0}, ψ, f). Then, if v_{∞} is the unique solution of P_{∞}(u_{0}, ψ, f), we have

lim_{ε→0} sup_{t∈[0,T]} ∥u_{∞,ε}(t, ·) − v_{∞}(t, ·)∥_{L^{2}(Ω)} = 0.

Closely related to the present work are the references [7] and [8] where the homogeneous Neumann problem and its limit as p goes to infinity or to one are considered. The difference here is that we are now considering Dirichlet boundary conditions, not only the homogeneous case, but also the non homogeneous case, and this introduces new difficulties specially when one tries to recover the local models when ε → 0. Remark that in our nonlocal formulation we are not imposing any continuity between the values of u inside Ω and outside it, ψ. However, when dealing with local problems usually the boundary datum is taken in the sense of traces, that is u|_{∂Ω} = ψ. Recovering this condition as ε → 0 is one of the main contributions of the present work.

Note that, as it happens for the local p−Laplacian, the Dirichlet problem can be written as a Neumann problem with a particular flux that depends on the solution itself. Indeed, the problem P_{p}^{J}(u_{0}, ψ) can be written as

\{ \begin{matrix}u_{t}(t, x) = ∫_{Ω} J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x)) dy + φ(x, u(x)) \\ (t, x) ∈]0, T [×Ω, \\ u(0, x) = u_{0}(x), x ∈ Ω, \end{matrix}

where

φ(x, u(x)) = ∫_{Ω_{J} \\overline{Ω}} J(x − y)|ψ(t, y) − u(t, x)|^{p−2}(ψ(t, y) − u(t, x)) dy.

In the homogeneous case, ψ ≡ 0,

φ(x, u(x)) = − (∫_{Ω_{J} \\overline{Ω}} J(x − y) dy) |u(t, x)|^{p−2}u(t, x).

This problem is a nonhomogeneous Neumann problem (see [7]) with a prescribed flux given by φ.

Let us finish the introduction by collecting some notations and results that will be used in the sequel. Following [9] (see also [2]), let

(1.4)  X(Ω) = {z ∈ L^{∞}(Ω, ℝ^{n}) : div(z) ∈ L^{1}(Ω)} .

If z ∈ X(Ω) and w ∈ BV (Ω) ∩ L^{∞}(Ω) we define the functional (z, Dw) : C_{0}^{∞}(Ω) → ℝ by the formula

(1.5)  < (z, Dw), φ >= − ∫_{Ω} w φ div(z) dx − ∫_{Ω} w z · ∇φ dx.

Then (z, Dw) is a Radon measure in Ω,

(1.6)  ∫_{Ω}(z, Dw) = ∫_{Ω} z · ∇w dx

for all w ∈ W^{1,1}(Ω) ∩ L^{∞}(Ω) and

(1.7)  | ∫_{B}(z, Dw)| ≤ ∫_{B} |(z, Dw)| ≤ ∥z∥_{∞} ∫_{B} ∥Dw∥

for any Borel set B ⊆ Ω.

In [9], a weak trace on ∂Ω of the normal component of z ∈ X(Ω) is defined. Concretely, it is proved that there exists a linear operator γ : X(Ω) → L^{∞}(∂Ω) such that

∥γ(z)∥_{∞} ≤ ∥z∥_{∞},

and

γ(z)(x) = z(x) · ν(x) for all x ∈ ∂Ω if z ∈ C^{1}(\overline{Ω}, ℝ^{N}).

We shall denote γ(z)(x) by [z, ν](x). Moreover, the following Green’s formula, relating the function [z, ν] and the measure (z, Dw), for z ∈ X(Ω) and w ∈ BV (Ω) ∩ L^{∞}(Ω), is established

(1.8)  ∫_{Ω} w div(z) dx + ∫_{Ω}(z, Dw) = ∫_{∂Ω}[z, ν]w dH^{N−1}.

Organization of the paper. The rest of the paper is organized as follows. In the second section we prove the existence and uniqueness of strong solutions for the nonlocal p-Laplacian problem with Dirichlet boundary conditions for p > 1 and we show that our model approaches local p−Laplacian evolution equation with Dirichlet boundary condition. In Section 3 we study the Dirichlet problem for the non local total variation flow, proving convergence to the local model when the problem is rescaled appropriately as well. Finally, in Section 4 we study the case p = ∞, obtaining a model for sandpiles with Dirichlet boundary conditions.

2. The case p > 1

2.1. Existence of solutions for the nonlocal problems. We first study P_{p}^{J}(u_{0}, ψ) from the point of view of Nonlinear Semigroup Theory. For that we introduce in L^{1}(Ω) the following operator associated with our problem.

Definition 2.1. For 1 < p < +∞ and ψ : Ω_{J} \ \overline{Ω} → ℝ, such that |ψ|^{p−1} ∈ L^{1}(Ω_{J} \ \overline{Ω}), we define in L^{1}(Ω) the operator B_{p,ψ}^{J} by

\begin{aligned}B_{p,ψ}^{J}(u)(x) & = − ∫_{Ω} J(x − y)|u(y) − u(x)|^{p−2}(u(y) − u(x)) dy \\ & − ∫_{Ω_{J} \\overline{Ω}} J(x − y)|ψ(y) − u(x)|^{p−2}(ψ(y) − u(x)) dy, x ∈ Ω. \end{aligned}

Remark 2.2. (i). We will set overall the section,

u_{ψ}(x) := \{ \begin{matrix}u(x) & if x ∈ Ω, \\ ψ(x) & if x ∈ Ω_{J} \ Ω. \end{matrix}

Therefore, we can rewrite

B_{p,ψ}^{J}(u)(x) = − ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p−2}(u_{ψ}(y) − u(x)) dy, x ∈ Ω.

(ii) If ψ = 0, then

\begin{matrix}B_{p,0}^{J}(u)(x) = − ∫_{Ω} J(x − y)|u(y) − u(x)|^{p−2}(u(y) − u(x)) dy \\ + (∫_{Ω_{J} \Ω} J(x − y)dy) |u(x)|^{p−2}u(x) , x ∈ Ω. \end{matrix}

Remark 2.3. It is easy to see that, (i) If ψ = 0, B_{p,0}^{J} is positively homogeneous of degree p − 1, (ii) L^{p−1}(Ω) ⊂ Dom(B_{p,ψ}^{J}), if p > 2. (iii) For 1 < p ≤ 2, Dom(B_{p,ψ}^{J}) = L^{1}(Ω) and B_{p,ψ}^{J} is closed in L^{1}(Ω) × L^{1}(Ω).

We have the following monotonicity lemma, whose proof is straightforward.

Lemma 2.4. Let 1 < p < +∞, ψ : Ω_{J} \ \overline{Ω} → ℝ, |ψ|^{p−1} ∈ L^{1}(Ω_{J} \ \overline{Ω}), and T : ℝ → ℝ a nondecreasing function. Then,

(i) for every u, v ∈ L^{p}(Ω) such that T (u − v) ∈ L^{p}(Ω), it holds

(2.1) \begin{matrix}∫_{Ω}(B_{p,ψ}^{J}u(x) − B_{p,ψ}^{J}v(x))T (u(x) − v(x))dx = \\ \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y) (T (u_{ψ}(y) − v_{ψ}(y)) − T (u_{ψ}(x) − v_{ψ}(x))) × \\ × (|u_{ψ}(y) − u_{ψ}(x)|^{p−2}(u_{ψ}(y) − u_{ψ}(x)) − |v_{ψ}(y) − v_{ψ}(x)|^{p−2}(v_{ψ}(y) − v_{ψ}(x))) dy dx. \end{matrix}

(ii) Moreover, if T is bounded, (2.1) holds for u, v ∈ Dom(B_{p,ψ}^{J}).

We have the following Poincaré’s type inequality.

Proposition 2.5. Given Ω and J as before, p ≥ 1 and ψ ∈ L^{p}(Ω_{J} \ \overline{Ω}), there exists λ = λ(J, Ω, p) > 0 such that

(2.2)  λ ∫_{Ω} |u(x)|^{p} dx ≤ ∫_{Ω} ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx + ∫_{Ω_{J} \\overline{Ω}} |ψ(y)|^{p} dy

for all u ∈ L^{p}(Ω).

Proof. Let r > 0 such that β ≥ J(x) ≥ α > 0 in B(0, r). Let

B_{0} = \{x ∈ Ω_{J} \ \overline{Ω} : d(x, Ω) ≤ r/2\},
B_{1} = {x ∈ Ω : d(x, B_{0}) ≤ r/2},
B_{j} = {x ∈ Ω \ ∪_{k=1}^{j−1}B_{k} : d(x, B_{j−1}) ≤ r/2} , j = 2, 3, ...

Observe that we can cover Ω by a finite number of non null sets {B_{j}}_{j=1}^{l_{r}}. Now

∫_{Ω} ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx ≥ ∫_{B_{j}} ∫_{B_{j−1}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx, j = 1, ..., l_{r},

and

∫_{B_{j}} ∫_{B_{j−1}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx

≥ \frac{1}{2^{p}} ∫_{B_{j}} ∫_{B_{j−1}} J(x − y)|u(x)|^{p} dy dx − ∫_{B_{j}} ∫_{B_{j−1}} J(x − y)|u_{ψ}(y)|^{p} dy dx

= \frac{1}{2^{p}} ∫_{B_{j}} (∫_{B_{j−1}} J(x − y) dy) |u(x)|^{p} dx − ∫_{B_{j−1}} (∫_{B_{j}} J(x − y) dx) |u_{ψ}(y)|^{p} dy

≥ \frac{1}{2^{p}} α|B_{j−1}| ∫_{B_{j}} |u(x)|^{p} dx − β|B_{j}| ∫_{B_{j−1}} |u_{ψ}(y)|^{p} dy.

Hence

\begin{matrix}∫_{Ω} ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx \\ ≥ \frac{1}{2^{p}} α|B_{j−1}| ∫_{B_{j}} |u(x)|^{p} dx − β|B_{j}| ∫_{B_{j−1}} |u_{ψ}(y)|^{p} dy. \end{matrix}

Therefore, since u_{ψ}(y) = ψ(y) if y ∈ B_{0}, u_{ψ}(y) = u(y) if y ∈ B_{j}, j = 1, …, l_{r}, B_{j} ∩ B_{i} = ∅, for all i ≠ j and |Ω \ ∪_{j=1}^{j_{r}}B_{j}| = 0, it is easy to see that there exists λ̂ = λ̂(J, Ω, p) > 0 such that

∫_{Ω} |u|^{p} ≤ λ̂ ∫_{Ω} ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p} dy dx + λ̂ ∫_{B_{0}} |ψ|^{p}.

The proof is finished by taking λ = λ̂^{−1}. □

In the next result we prove that B_{p,ψ}^{J} is a completely accretive operator (see [16]) and verifies a range condition. In short, this means that for any ϕ ∈ L^{p}(Ω) there is a unique solution of the problem u + B_{p,ψ}^{J}u = ϕ and the resolvent (I + B_{p,ψ}^{J})^{−1} is a contraction in L^{q}(Ω) for all 1 ≤ q ≤ +∞.

Theorem 2.6. Let 1 < p < +∞. For ψ ∈ L^{p}(Ω_{J} \ \overline{Ω}), the operator B_{p,ψ}^{J} is completely accretive and verifies the range condition

(2.3)  L^{p}(Ω) ⊂ Ran(I + B_{p,ψ}^{J}).

Proof. Given u_{i} ∈ Dom(B_{p,ψ}^{J}), i = 1, 2, by the monotonicity Lemma 2.4, for any q ∈ C^{∞}(ℝ), 0 ≤ q′ ≤ 1, supp(q′) compact, 0 ∉ supp(q), we have that

∫_{Ω}(B_{p,ψ}^{J}u_{1}(x) − B_{p,ψ}^{J}u_{2}(x))q(u_{1}(x) − u_{2}(x)) dx ≥ 0,

from where it follows that B_{p,ψ}^{J} is a completely accretive operator (see [16]).

To show that B_{p,ψ}^{J} satisfies the range condition we have to prove that for any ϕ ∈ L^{p}(Ω) there exists u ∈ Dom(B_{p,ψ}^{J}) such that ϕ = u + B_{p,ψ}^{J}u.

Assume first p ≥ 2. Let ϕ ∈ L^{p}(Ω) and set

K = \{w ∈ L^{p}(Ω_{J}) : w = ψ in Ω_{J} \ \overline{Ω}\} .

We consider the continuous monotone operator A : K → L^{p′}(Ω_{J}) defined by

A(w)(x) := w(x) − ∫_{Ω_{J}} J(x − y)|w(y) − w(x)|^{p−2}(w(y) − w(x)) dy.

A is coercive in L^{p}(Ω_{J}). In fact, by Proposition 2.5, for any w ∈ K,

∫_{Ω_{J}} A(w)w = ∫_{Ω_{J}} w^{2} − ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|w(y) − w(x)|^{p−2}(w(y) − w(x)) dyw(x)dx

≥ \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|w(y) − w(x)|^{p} dydx

≥ \frac{1}{2} ∫_{Ω} ∫_{Ω_{J}} J(x − y)|w_{ψ}(y) − w(x)|^{p} dydx ≥ \frac{λ}{2}∥w∥_{L^{p}(Ω)}^{p} − \frac{1}{2} ∫_{Ω_{J} \\overline{Ω}} |ψ|^{p}.

Therefore

lim ; \begin{matrix}∥w∥_{L^{p}(Ω_{J})} → +∞ \\ w ∈ K \end{matrix} \frac{∫_{Ω_{J}} A(w)w}{∥w∥_{L^{p}(Ω_{J})}} = +∞.

Now, since p ≥ 2, we have the function ϕ_{ψ} ∈ L^{p′}(Ω_{J}). Then, applying [33, Corollary III.1.8.] to the operator B(w) := A(w) − ϕ_{ψ}, we get there exists w ∈ K, such that

w(x) − ∫_{Ω_{J}} J(x − y)|w(y) − w(x)|^{p−2}(w(y) − w(x)) dy = ϕ_{ψ}(x) for all x ∈ Ω_{J}.

Hence, u := w_{|Ω} satisfies

u(x) − ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u(x)|^{p−2}(u_{ψ}(y) − u(x)) dy = ϕ(x) for all x ∈ Ω,

and consequently, ϕ = u + B_{p,ψ}^{J}u.

Suppose now 1 < p < 2. By the results in [7], we know that the operator

B_{p}^{J}u(x) = − ∫_{Ω} J(x − y)|u(y) − u(x)|^{p−2}(u(y) − u(x)) dy

is m-accretive in L^{1}(Ω) and satisfies what is called property (M_{0}), that is, for any q ∈ C^{∞}(ℝ), 0 ≤ q′ ≤ 1, supp(q′) compact, 0 ∉ supp(q), and (u, v) ∈ B_{p}^{J},

∫_{Ω} q(u)v ≥ 0.

On the other hand,

φ(x, r) = − ∫_{Ω_{J} \Ω} J(x − y)|ψ(y) − r|^{p−2}(ψ(y) − r) dy

is continuous and non-decreasing in r for almost every x ∈ Ω, and an L^{1}(Ω)-function for all r. Therefore, by [3, Theorem 3.1], B_{p,ψ}^{J}u(x) = B_{p}^{J}u(x) + φ(x, u(x)) is m-accretive in L^{1}(Ω). □

If B_{p,ψ}^{J} denotes the closure of B_{p,ψ}^{J} in L^{1}(Ω), by Theorem 2.6, we have B_{p,ψ}^{J} is m-completely accretive in L^{1}(Ω).

Next we get the following result, from which Theorem 1.2 can be derived.

Theorem 2.7. Assume p > 1. Let T > 0, ψ ∈ L^{p}(Ω_{J} \ \overline{Ω}) and u_{0} ∈ L^{1}(Ω). Then, there exists a unique mild solution u of

(2.4)  \{ \begin{matrix}u′(t) + B_{p,ψ}^{J}u(t) = 0,  t ∈ (0, T), \\ u(0) = u_{0}. \end{matrix}

Moreover,

(1) if u_{0} ∈ L^{p}(Ω), the unique mild solution u of (2.4) is a solution of P_{p}^{J}(u_{0}, ψ) in the sense of Definition 1.1. If 1 < p ≤ 2, this is true for any u_{0} ∈ L^{1}(Ω) and any ψ such that |ψ|^{p−1} ∈ L^{1}(Ω_{J} \ \overline{Ω}).

(2) Let u_{i}_{0} ∈ L^{1}(Ω) and u_{i} a solution in [0, T] of P_{p}^{J}(u_{i}_{0}), i = 1, 2. Then

∫_{Ω}(u_{1}(t) − u_{2}(t))^{+} ≤ ∫_{Ω}(u_{1}_{0} − u_{2}_{0})^{+} for every t ∈]0, T [.

Moreover, for q ∈ [1, +∞], if u_{i}_{0} ∈ L^{q}(Ω), i = 1, 2, then

∥u_{1}(t) − u_{2}(t)∥_{L^{q}(Ω)} ≤ ∥u_{1}_{0} − u_{2}_{0}∥_{L^{q}(Ω)} for every t ∈]0, T [.

Proof. As a consequence of Theorem 2.6 we get the existence of mild solution of (2.4) (see [17] and [16]). On the other hand, u(t) is a solution of P_{p}^{J}(u_{0}, ψ) if and only if u(t) is a strong solution of the abstract Cauchy problem (2.4). Now, due to the complete accretivity of B_{p,ψ}^{J} and the range condition (2.3), u(t) is a strong solution (see [16]). Moreover, in the case 1 < p ≤ 2, since Dom(B_{p,ψ}^{J}) = L^{1}(Ω) and B_{p,ψ}^{J} is closed in L^{1}(Ω) × L^{1}(Ω), the result holds for L^{1}-data. Finally, the contraction principle is a consequence of the general Nonlinear Semigroup Theory ([17], [29]). □

2.2. Convergence to the p-laplacian. Our main goal in this section is to show that the solution to the Dirichlet problem for the p-Laplacian equation, D_{p}(u_{0}, ψ̃), can be approximated by solutions to suitable nonlocal Dirichlet problems P_{p}^{J}(u_{0}, ψ).

Let us first recall the following result from [7].

Proposition 2.8 ([7]). Let 1 ≤ q < +∞, D a bounded domain in ℝ^{N}, ρ : ℝ^{N} → ℝ a nonnegative continuous radial function with compact support, non identically zero, and ρ_{n}(x) := n^{N}ρ(nx). Let {f_{n}} be a sequence of functions in L^{q}(D) such that

(2.5)  ∫_{D} ∫_{D} |f_{n}(y) − f_{n}(x)|^{q}ρ_{n}(y − x) dx dy ≤ M \frac{1}{n^{q}} .

1. If {f_{n}} is weakly convergent in L^{q}(D) to f then (i) if q > 1, f ∈ W^{1,q}(D) and moreover

(ρ(z))^{1/q} χ_{D} (x + \frac{1}{n}z) \frac{f_{n} (x + \frac{1}{n}z) − f_{n}(x)}{1/n} ⇀ (ρ(z))^{1/q} z · ∇f

weakly in L^{q}(D) × L^{q}(ℝ^{N}); (ii) if q = 1, f ∈ BV (D) and moreover

ρ(z)χ_{D} (x + \frac{1}{n}z) \frac{f_{n} (x + \frac{1}{n}z) − f_{n}(x)}{1/n} ⇀ ρ(z)z · ∇f

weakly as measures.

2. Assume D is a smooth bounded domain in ℝ^{N} and ρ(x) ≥ ρ(y) if |x| ≤ |y|. Then {f_{n}} is relatively compact in L^{q}(D) and, consequently, there exists a subsequence {f_{n_{k}}} such that

(i) if q > 1, f_{n_{k}} → f in L^{q}(D) with f ∈ W^{1,q}(D);

(ii) If q = 1, f_{n_{k}} → f in L^{1}(D) with f ∈ BV (D).

Let us now recall some results about the p-Laplacian equation

D_{p}(u_{0}, ψ̃) \{ \begin{matrix}u_{t} = Δ_{p}u & in ]0, T [×Ω, \\ u = ψ̃ & on ]0, T [×∂Ω, \\ u(0, x) = u_{0}(x) & in Ω. \end{matrix}

In the case ψ̃ ∈ W^{1/p′,p}(∂Ω), associated to the p-Laplacian with non homogeneous Dirichlet boundary condition, in [2] it is defined the operator A_{p,ψ} ⊂ L^{1}(Ω) × L^{1}(Ω) as (u, û) ∈ A_{p,ψ̃}

if and only if û ∈ L^{1}(Ω), u ∈ W 1,p ; ψ̃(Ω) := {u ∈ W^{1,p}(Ω) : u|_{∂Ω} = ψ̃ H^{N−1} − a.e. on ∂Ω} and

∫_{Ω} |∇u|^{p−2}∇u · ∇(u − v) ≤ ∫_{Ω} û(u − v) for every v ∈ W 1,p ; ψ̃(Ω) ∩ L^{∞}(Ω).

This inequality is equivalent to

∫_{Ω} |∇u|^{p−2}∇u · ∇w = ∫_{Ω} ûw for every w ∈ W_{0}^{1,p}(Ω) ∩ L^{∞}(Ω).

Moreover, for ψ̃ ∈ W^{1/p′,p}(∂Ω)∩L^{∞}(∂Ω), A_{p,ψ̃} is proved to be a completely accretive operator in L^{1}(Ω) satisfying the range condition L^{∞}(Ω) ⊂ Ran(I + A_{p,ψ̃}), and it is easy to see that

\overline{D(A_{p,ψ̃})}^{L^{1}(Ω)} = L^{1}(Ω). Therefore, its closure A_{p,ψ̃} in L^{1}(Ω) × L^{1}(Ω) is an m-completely accretive operator in L^{1}(Ω). Consequently, for any u_{0} ∈ L^{1}(Ω) there exists a unique mildsolution u(t) = e^{−tA_{p,ψ̃}}u_{0} of the abstract Cauchy problem associated to D_{p}(u_{0}, ψ̃), given by the Crandall-Liggett’s exponential formula. Due to the complete accretivety of the operator A_{p,ψ̃}, in the case u_{0} ∈ D(A_{p,ψ̃}) this mild solution is the unique strong solution of problem D_{p}(u_{0}, ψ̃).

In the homogeneous case ψ̃ = 0, due to the results in [15], we can say that for any u_{0} ∈ L^{1}(Ω), the mild-solution u(t) = e^{−tA_{p,0}}u_{0} is the unique entropy solution of problem D_{p}(u_{0}, 0).

For given p > 1 and J, we consider the rescaled kernels

J_{p,ε}(x) := \frac{C_{J,p}}{ε^{p+N}} J (\frac{x}{ε}) ,  where  C_{J,p}^{−1} := \frac{1}{2} ∫_{ℝ^{N}} J(z)|z_{N}|^{p} dz

is a normalizing constant in order to obtain the p-Laplacian in the limit instead of a multiple of it.

Proposition 2.9. Let Ω be a smooth bounded domain in ℝ^{N} and ψ̃ ∈ W^{1/p′,p}(∂Ω)∩L^{∞}(∂Ω). Let ψ ∈ W^{1,p}(Ω_{J}) ∩ L^{∞}(Ω_{J}) such that ψ|_{∂Ω} = ψ̃. Assume J(x) ≥ J(y) if |x| ≤ |y|. Then, for any ϕ ∈ L^{∞}(Ω),

(2.6)  (I + B_{p,ψ}^{J_{p,ε}})^{−1} ϕ → (I + A_{p,ψ̃})^{−1} ϕ in L^{p}(Ω) as ε → 0.

Proof. We denote

Ω_{ε} := Ω_{J_{p,ε}} = Ω + supp(J_{p,ε}).

For ε > 0 small, let u_{ε} = (I + B_{p,ψ}^{J_{p,ε}})^{−1} ϕ. Then,

(2.7)  \begin{aligned}_{∫Ω}u_{ε}v − \frac{C_{J,p}}{ε^{p+N}} & _{∫Ω}_{∫Ω_{ε}}J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − u_{ε}(x)|^{p−2}× \\ & ×((u_{ε})_{ψ}(y) − u_{ε}(x)) dy v(x) dx = _{∫Ω}ϕv \end{aligned}

for every v ∈ L^{∞}(Ω).

Let M := max{∥ϕ∥_{L^{∞}(Ω)}, ∥ψ∥_{L^{∞}(Ω_{J})}}. Taking v = (u_{ε} − M)^{+} in (2.7), we get

∫_{Ω} u_{ε}(x)(u_{ε}(x) − M)^{+}dx

−\frac{C_{J,p}}{ε^{p+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − u_{ε}(x)|^{p−2}((u_{ε})_{ψ}(y) − u_{ε}(x)) dy (u_{ε}(x) − M)^{+} dx

= ∫_{Ω} ϕ(x)(u_{ε}(x) − M)^{+} dx.

Now,

−\frac{C_{J,p}}{ε^{p+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − u_{ε}(x)|^{p−2}((u_{ε})_{ψ}(y) − u_{ε}(x)) dy (u_{ε}(x) − M)^{+}dx

= −\frac{C_{J,p}}{ε^{p+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − u_{ε}(x)|^{p−2}((u_{ε})_{ψ}(y) − u_{ε}(x)) dy ((u_{ε})_{ψ}(x) − M)^{+}dx

= \frac{C_{J,p}}{2ε^{p+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − u_{ε}(x)|^{p−2}((u_{ε})_{ψ}(y) − u_{ε}(x))

×(((u_{ε})_{ψ}(y) − M)^{+} − ((u_{ε})_{ψ}(x) − M)^{+}) dy dx ≥ 0.

Therefore

∫_{Ω} u_{ε}(x)(u_{ε}(x) − M)^{+}dx ≤ ∫_{Ω} ϕ(x)(u_{ε}(x) − M)^{+} dx.

Consequently, we have

∫_{Ω}(u_{ε}(x) − M)(u_{ε}(x) − M)^{+}dx ≤ ∫_{Ω}(ϕ(x) − M)(u_{ε}(x) − M)^{+} dx ≤ 0,

and u_{ε}(x) ≤ M for almost all x ∈ Ω. Analogously, we can obtain −M ≤ u_{ε}(x) for almost all x ∈ Ω. Thus

(2.8)  ∥u_{ε}∥_{L^{∞}(Ω)} ≤ M for all ϵ > 0,

and therefore, there exists a sequence ε_{n} → 0 such that

u_{ε_{n}} ⇀ u weakly in L^{1}(Ω).

Taking v = u_{ε} − ψ in (2.7) we get

(2.9)  \begin{matrix}_{∫Ω}u_{ε}(u_{ε} − ψ) − \frac{C_{J,p}}{ε^{p+N}} _{∫Ω_{ε}}_{∫Ω_{ε}}J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)|^{p−2}× \\ ×((u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)) dy ((u_{ε})_{ψ}(x) − ψ(x)) dx = _{∫Ω}ϕ(u_{ε} − ψ) \end{matrix}

Now, by (2.9) and (2.8),

\begin{matrix}\frac{C_{J,p}}{2ε^{N}} ∫_{Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) \frac{|(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)|^{p}}{ε^{p}} dy dx \\ ≤ \frac{C_{J,p}}{2ε^{N}} ∫_{Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) \frac{|(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)|^{p−1}}{ε^{p−1}} \frac{|ψ(y) − ψ(x)|}{ε}dy dx + M_{1}. \end{matrix}

Since ψ̃ ∈ W^{1,p}(Ω_{J}), using Young’s inequality, we obtain

\frac{1}{ε^{N}} ∫_{Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) \frac{|(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)|^{p}}{ε^{p}} dy dx ≤ M_{2}.

Moreover,

(2.10)

∫_{Ω_{J}} ∫_{Ω_{J}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)}{ε}|^{p} dx dy

= ∫_{Ω_{ε}} ∫_{Ω_{ε}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)}{ε}|^{p} dx dy

+2 ∫_{Ω_{J} \Ω_{ε}} ∫_{Ω_{ε}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{ψ(y) − (u_{ε})_{ψ}(x)}{ε}|^{p} dx dy

+ ∫_{Ω_{J} \Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{ψ(y) − ψ(x)}{ε}|^{p} dx dy

= ∫_{Ω_{ε}} ∫_{Ω_{ε}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)}{ε}|^{p} dx dy

+2 ∫_{Ω_{J} \Ω_{ε}} ∫_{Ω_{ε}\Ω} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{ψ(y) − ψ(x)}{ε}|^{p} dx dy

+ ∫_{Ω_{J} \Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} \frac{1}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{ψ(y) − ψ(x)}{ε}|^{p} dx dy ≤ M_{3}.

Therefore, by Proposition 2.8, there exists a subsequence, denoted as above, and w ∈ W^{1,p}(Ω_{J}) such that

(u_{ε_{n}})_{ψ} → w strongly in L^{p}(Ω_{J}),

hence w = u in Ω, u ∈ W 1,p ; ψ̃(Ω), and

(2.11) (\frac{C_{J,p}}{2} J(z))^{1/p} χ_{Ω}(x + ε_{n}z)\frac{u_{ε_{n}}(x + ε_{n}z) − u_{ε_{n}}(x)}{ε_{n}} ⇀ (\frac{C_{J,p}}{2} J(z))^{1/p} z · ∇u(x)

weakly in L^{p}(Ω) × L^{p}(ℝ^{N}). We can also assume that

(J(z))^{1/p′} |\frac{(u_{ε_{n}})_{ψ}(x + ε_{n}z) − (u_{ε_{n}})_{ψ}(x)}{ε_{n}} |^{p−2} χ_{Ω_{εn}}(x + ε_{n}z)\frac{(u_{ε_{n}})_{ψ}(x + ε_{n}z) − (u_{ε_{n}})_{ψ}(x)}{ε_{n}}

⇀ (J(z))^{1/p′} χ(x, z)

weakly in L^{p′}(Ω_{J}) × L^{p′}(ℝ^{N}).

Passing to the limit in (2.7) for ε = ε_{n}, we get

(2.12)  ∫_{Ω} uv + ∫_{ℝ^{N}} ∫_{Ω} \frac{C_{J,p}}{2} J(z)χ(x, z) z · ∇v(x) dx dz = _{∫Ω}ϕv

for every v smooth with support in Ω and by approximation for every v ∈ W_{0}^{1,p}(Ω).

Finally, working as in Proposition 3.3. of [7], we can prove

(2.13)  ∫_{ℝ^{N}} ∫_{Ω} \frac{C_{J,p}}{2} J(z)χ(x, z)z · ∇v(x) dx dz = ∫_{Ω} |∇u|^{p−2} ∇u · ∇v

and the proof is finished. □

From the above Proposition, by standard results of the Nonlinear Semigroup Theory (see,

[20] or [17]), we obtain Theorem 1.3.

3. The nonlocal Total Variation Flow. The case p = 1

3.1. Existence of solutions for the nonlocal problem. This section deals with the existence and uniqueness of solutions for the nonlocal 1-Laplacian problem with Dirichlet boundary condition,

P_{1}^{J}(u_{0}, ψ) \{ \begin{matrix}u_{t}(t, x) = ∫_{Ω} J(x − y)\frac{u(t, y) − u(t, x)}{|u(t, y) − u(t, x)|} dy. \\ + ∫_{Ω_{J} \\overline{Ω}} J(x − y)\frac{ψ(y) − u(t, x)}{|ψ(y) − u(t, x)|} dy, x ∈ Ω. \\ u(0, x) = u_{0}(x). \end{matrix}

As in the case p > 1, to prove existence and uniqueness of solutions of P_{1}^{J}(u_{0}, ψ) we use the Nonlinear Semigroup Theory, so we start by introducing the following operator in L^{1}(Ω).

Definition 3.1. Given ψ ∈ L^{1}(Ω_{J} \ \overline{Ω}), we define the operator B_{1,ψ}^{J} in L^{1}(Ω) × L^{1}(Ω) by û ∈ B_{1,ψ}^{J}u if and only if u, û ∈ L^{1}(Ω), there exists g ∈ L^{∞}(Ω_{J} × Ω_{J}), g(x, y) = −g(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, ∥g∥_{∞} ≤ 1,

(3.1)  û(x) = − ∫_{Ω_{J}} J(x − y)g(x, y) dy a.e. x ∈ Ω,

and

(3.2)  J(x − y)g(x, y) ∈ J(x − y) sign(u(y) − u(x)) a.e. (x, y) ∈ Ω × Ω,
(3.3)  J(x − y)g(x, y) ∈ J(x − y) sign(ψ(y) − u(x)) a.e. (x, y) ∈ Ω × (Ω_{J} \ Ω).

Remark 3.2. Observe that (i) we can rewrite (3.2) + (3.3) as

(3.4)  J(x − y)g(x, y) ∈ J(x − y) sign(u_{ψ}(y) − u(x)) a.e. (x, y) ∈ Ω × Ω_{J},

where we set as above, and overall the section,

u_{ψ}(x) := \{ \begin{matrix}u(x) & if x ∈ Ω, \\ ψ(x) & if x ∈ Ω_{J} \ Ω. \end{matrix}

(ii) It holds L^{1}(Ω) = Dom(B_{1,ψ}^{J}) and B_{1,ψ}^{J} is closed in L^{1}(Ω) × L^{1}(Ω).

(iii) It is not difficult to see that, if g ∈ L^{∞}(Ω_{J} × Ω_{J}), g(x, y) = −g(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, ∥g∥_{∞} ≤ 1,

J(x − y)g(x, y) ∈ J(x − y) sign(z(y) − z(x)) a.e. (x, y) ∈ Ω_{J} × Ω_{J},

is equivalent to

− ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)g(x, y) dy z(x) dx = \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|z(y) − z(x)| dy dx,

Theorem 3.3. Let ψ ∈ L^{1}(Ω_{J} \ \overline{Ω}). The operator B_{1,ψ}^{J} is completely accretive and satisfies the range condition

L^{∞}(Ω) ⊂ Ran(I + B_{1,ψ}^{J}).

Proof. Let û_{i} ∈ B_{1,ψ}^{J}u_{i}, i = 1, 2, and set u_{i}(y) = ψ(y) in Ω_{J} \ Ω. Then, there exist g_{i} ∈ L^{∞}(Ω_{J} × Ω_{J}), ∥g_{i}∥_{∞} ≤ 1, g_{i}(x, y) = −g_{i}(y, x), J(x − y)g_{i}(x, y) ∈ J(x − y)sign(u_{i}(y) − u_{i}(x)) for almost all (x, y) ∈ Ω × Ω_{J}, such that,

û_{i}(x) = − ∫_{Ω_{J}} J(x − y)g_{i}(x, y) dy a.e. x ∈ Ω,

for i = 1, 2. Given q ∈ C^{∞}(ℝ), 0 ≤ q′ ≤ 1, supp(q′) compact, 0 ∉ supp(q), we have

∫_{Ω}(û_{1}(x) − û_{2}(x))q(u_{1}(x) − u_{2}(x)) dx

= \frac{1}{2} ∫_{Ω} ∫_{Ω} J(x − y)(g_{1}(x, y) − g_{2}(x, y)) (q(u_{1}(y) − u_{2}(y)) − q(u_{1}(x) − u_{2}(x))) dxdy

− ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y)(g_{1}(x, y) − g_{2}(x, y) (q(u_{1}(x) − u_{2}(x))) dx dy

≥ \frac{1}{2} ∫_{Ω} ∫_{Ω} J(x − y)(g_{1}(x, y) − g_{2}(x, y)) (q(u_{1}(y) − u_{2}(y)) − q(u_{1}(x) − u_{2}(x))) dxdy

Now, by the mean value Theorem

J(x − y)(g_{1}(x, y) − g_{2}(x, y)) [q(u_{1}(y) − u_{2}(y)) − q(u_{1}(x) − u_{2}(x))]
= J(x − y)(g_{1}(x, y) − g_{2}(x, y))q′(ξ) [(u_{1}(y) − u_{2}(y)) − (u_{1}(x) − u_{2}(x))]
= J(x − y)q′(ξ) [g_{1}(x, y)(u_{1}(y) − u_{1}(x)) − g_{1}(x, y)(u_{2}(y) − u_{2}(x))]
−J(x − y)q′(ξ) [g_{2}(x, y)(u_{1}(y) − u_{1}(x)) − g_{1}(x, y)(u_{2}(y) − u_{2}(x))] ≥ 0,

since

J(x − y)g_{i}(x, y)(u_{i}(y) − u_{i}(x)) = J(x − y)|u_{i}(y) − u_{i}(x)|, i = 1, 2,

and

−J(x − y)g_{i}(x, y)(u_{j}(y) − u_{j}(x)) ≥ −J(x − y)|u_{j}(y) − u_{j}(x)|, i ≠ j.

Hence

∫_{Ω}(û_{1}(x) − û_{2}(x))q(u_{1}(x) − u_{2}(x)) dx ≥ 0,

from where it follows that B_{1,ψ}^{J} is a completely accretive operator.

To show that B_{1,ψ}^{J} satisfies the range condition, let us see that for any ϕ ∈ L^{∞}(Ω),

lim_{p→1+}(I + B_{p,ψ}^{J})^{−1}ϕ = (I + B_{1,ψ}^{J})^{−1}ϕ weakly in L^{1}(Ω).

We prove this in several steps. Step 1. Let us first suppose that ψ ∈ L^{∞}(Ω_{J} \ \overline{Ω}). For 1 < p < +∞, by Theorem 2.6, there is u_{p} such that u_{p} = (I + B_{p,ψ}^{J})^{−1}ϕ, that is,

(3.5)  u_{p}(x) −_{∫Ω_{J}} J (x − y) |(u_{p})_{ψ}(y) − u_{p}(x)|^{p−2}((u_{p})_{ψ}(y) − u_{p}(x)) dy = ϕ(x),

a.e. x ∈ Ω. It is easy to see that, ∥u_{p}∥_{∞} ≤ sup{∥ϕ∥_{∞}, ∥ψ∥_{∞}}. Therefore, there exists a sequence p_{n} → 1 such that

u_{p_{n}} ⇀ u weakly in L^{2}(Ω).

16 F. ANDREU, J. M. MAZÓN, J. D. ROSSI AND J. TOLEDO

On the other hand, we also have

\frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y) |(u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}} dy dx ≤ M_{2}, ∀ n ∈ ℕ.

Consequently, for any measurable subset E ⊂ Ω_{J} × Ω_{J}, we have

\begin{matrix}|∫ ∫_{E} J(x − y)|(u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−2} ((u_{p_{n}}(y))_{ψ} − (u_{p_{n}})_{ψ}(x)) | \\ ≤ ∫ ∫_{E} J(x − y)|(u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−1} ≤ M_{2}|E|^{\frac{1}{pn}} . \end{matrix}

Hence, by the Dunford-Pettis Theorem we may assume that there exists g(x, y) such that

J(x − y)|(u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−2} ((u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)) ⇀ J(x − y)g(x, y), weakly in L^{1}(Ω_{J} × Ω_{J}), g(x, y) = −g(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, and ∥g∥_{∞} ≤ 1.

Therefore, by (3.5),

(3.6)  u(x) − ∫_{Ω_{J}} J(x − y)g(x, y) dy = ϕ(x) a.e. x ∈ Ω.

Then, to finish the proof it is enough to show that

(3.7) − ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)g(x, y) dy u_{ψ}(x) dx = \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|u_{ψ}(y) − u_{ψ}(x)| dy dx.

In fact, by (3.5) and (3.6),

\frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y) |(u_{p_{n}})_{ψ}(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}} dy dx = ∫_{Ω} ϕu_{p_{n}} − ∫_{Ω} u_{p_{n}}u_{p_{n}}

− ∫_{Ω_{J} \Ω} ∫_{Ω_{J}} J(x − y) |ψ(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−2} (ψ(y) − (u_{p_{n}})_{ψ}(x)) dy ψ(x) dx = ∫_{Ω} ϕu − ∫_{Ω} uu − ∫_{Ω} ϕ(u − u_{p_{n}}) + ∫_{Ω} 2u(u − u_{p_{n}}) − ∫_{Ω}(u − u_{p_{n}})(u − u_{p_{n}})

− ∫_{Ω_{J} \Ω} ∫_{Ω_{J}} J(x − y) |ψ(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−2} (ψ(y) − (u_{p_{n}})_{ψ}(x)) dy ψ(x) dx

≤ − ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)g(x, y) dy u(x) dx − ∫_{Ω} ϕ(u − u_{p_{n}}) + ∫_{Ω} 2u(u − u_{p_{n}})

+ ∫_{Ω_{J} \Ω} ∫_{Ω_{J}} J(x − y)g(x, y) dy ψ(x) dx
− ∫_{Ω_{J} \Ω} ∫_{Ω_{J}} J(x − y) |ψ(y) − (u_{p_{n}})_{ψ}(x)|^{p_{n}−2} (ψ(y) − (u_{p_{n}})_{ψ}(x)) dy ψ(x) dx,

and so,

lim sup_{n→+∞} \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y) |u_{p_{n}}(y) − u_{p_{n}}(x)|^{p_{n}} dy dx ≤ − ∫_{Ω} ∫_{Ω} J(x − y)g(x, y) dy u(x) dx.

Now, by the monotonicity Lemma 2.4, for all ρ ∈ L^{∞}(Ω),

\begin{matrix}− ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|ρ(y) − ρ(x)|^{p_{n}−2}(ρ(y) − ρ(x)) dy (u_{p_{n}}(x) − ρ(x)) dx \\ ≤ − ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|u_{p_{n}}(y) − u_{p_{n}}(x)|^{p_{n}−2}(u_{p_{n}}(y) − u_{p_{n}}(x)) dy (u_{p_{n}}(x) − ρ(x)) dx. \end{matrix}

Taking limits,

− ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y) sign_{0}(ρ(y) − ρ(x)) dy (u(x) − ρ(x)) dx
≤ − ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)g(x, y) dy (u(x) − ρ(x)) dx.

Taking now, ρ = u ± λu, λ > 0, and letting λ → 0, we get (3.7), and the proof is finished for this class of data. Step 2. Let us now suppose that ψ^{−} is bounded. Let ψ_{n} = T_{n}(ψ), n large enough such that ψ_{n}^{−} = ψ^{−}. Then, {ψ_{n}} is a non decreasing sequence that converges in L^{1} to ψ. By Step 1, there exists u_{n} = (I +B_{1,ψ_{n}}^{J})^{−1}ϕ, that is, there exists g_{n} ∈ L^{∞}(Ω_{J} ×Ω_{J}), g_{n}(x, y) = −g_{n}(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, ∥g_{n}∥_{∞} ≤ 1,

(3.8)  u_{n}(x) − ∫_{Ω_{J}} J(x − y)g_{n}(x, y) dy = ϕ(x) a.e. x ∈ Ω.

and

(3.9)  \begin{matrix}− ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)g_{n}(x, y) dy (u_{n})_{ψ_{n}}(x) dx \\ = \frac{1}{2} ∫_{Ω_{J}} ∫_{Ω_{J}} J(x − y)|(u_{n})_{ψ_{n}}(y) − (u_{n})_{ψ_{n}}(x)| dy dx. \end{matrix}

Therefore, by monotonicity,

∫_{Ω_{J}} ∫_{Ω_{J}} ((u_{n})_{ψ_{n}} − (u_{n+1})_{ψ_{n+1}}) ((u_{n})_{ψ_{n}} − (u_{n+1})_{ψ_{n+1}})^{+} ≤ 0,

which implies u_{n} ≤ u_{n+1}. Since {u_{n}} is bounded in L^{∞} we have {u_{n}} converges to a function u in L^{2}. On the other hand, we can suppose that J(x − y)g_{n}(x, y) converges weakly in L^{2} to J(x − y)g(x, y), g(x, y) = −g(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, and ∥g∥_{∞} ≤ 1. Hence, passing to the limit in (3.8) and (3.9) we obtain u = (I + B_{1,ψ}^{J})^{−1}ϕ. Step 3. For a general ψ ∈ L^{1}(Ω_{J} \\overline{Ω}), apply Step 2 to ψ_{n} = sup\{ψ, −n\} and use monotonicity in a similar way to finish the proof. □

Proof of Theorem 1.5. As a consequence of the above results, we have that the abstract Cauchy problem

(3.10)  \{ \begin{matrix}u′(t) + B_{1,ψ}^{J}u(t) ∋ 0,  t ∈ (0, T), \\ u(0) = u_{0} \end{matrix}

has a unique mild solution u for every initial datum u_{0} ∈ L^{1}(Ω) and T > 0 (see [17]). Moreover, due to the complete accretivity of the operator B_{1,ψ}^{J}, the mild solution of (3.10) is a strong solution ([16]). Consequently, the the proof is concluded.

3.2. Convergence to the Total Variation Flow. Let us start recalling some results from [1] (see also [2]) about the Dirichlet problem for the total variational flow, that is,

D_{1}(u_{0}, ψ̃) \{ \begin{matrix}u_{t} = div (\frac{Du}{|Du|}) & in ]0, T [×Ω, \\ u = ψ̃ & on ]0, T [×∂Ω, \\ u(0, x) = u_{0}(x) & in Ω, \end{matrix}

with ψ̃ ∈ L^{1}(∂Ω).

Theorem 3.4 ([1]). Let T > 0 and ψ̃ ∈ L^{1}(∂Ω). For any u_{0} ∈ L^{1}(Ω) (L^{2}(Ω)) there exists a unique entropy (strong) solution u(t) of D_{1}(u_{0}, ψ̃).

Associated to −div (\frac{Du}{|Du|}) with Dirichlet boundary conditions, in [1] it is defined the operator Aψ̃ ⊂ L^{1}(Ω) × L^{1}(Ω) as follows, (u, v) ∈ Aψ̃ if and only if u, v ∈ L^{1}(Ω), q(u) ∈ BV (Ω) for all q ∈ P := {q ∈ W^{1,∞}(ℝ) : q′ ≥ 0, supp(q′) is compact}, and there exists ζ ∈ X(Ω), with ∥ζ∥_{∞} ≤ 1, v = −div(ζ) in D′(Ω) such that

(3.11)  ∫_{Ω}(w − q(u))v ≤ ∫_{Ω}(ζ, Dw) − |Dq(u)| + ∫_{∂Ω} |w − q(ψ̃))| − ∫_{∂Ω} |q(u) − q(ψ̃)|

for every w ∈ BV (Ω) ∩ L^{∞}(Ω) and every q ∈ P. Also in [1] it is proved that the following assertions are equivalent,

(a) (u, v) ∈ Aψ̃

(b) u, v ∈ L^{1}(Ω), q(u) ∈ BV (Ω) for all q ∈ P, and there exists ζ ∈ X(Ω), with ∥ζ∥_{∞} ≤ 1, v = −div(ζ) in D′(Ω) such that

(3.12)  ∫_{Ω}(ζ, Dq(u)) = |Dq(u)| ∀ q ∈ P

(3.13)  [ζ, ν] ∈ sign(q(ψ̃) − q(u)) H^{N−1} − a.e. on ∂Ω, ∀ q ∈ P.

Moreover, it is shown that Aψ̃ is an m-completely accretive operator in L^{1}(Ω) with dense domain and that for any u_{0} ∈ L^{1}(Ω), the unique entropy solution u(t) of problem D_{1}(u_{0}, ψ̃) coincides with the unique mild-solution e^{−tAψ̃}u_{0} given by the Crandall-Liggett’s exponential formula.

Now, given J, we consider the rescaled kernels

J_{1,ε}(x) := \frac{C_{J,1}}{ε^{1+N}} J (\frac{x}{ε}) ,  with  C_{J,1}^{−1} := \frac{1}{2} ∫_{ℝ^{N}} J(z)|z_{N}| dz

that is, a normalizing constant in order to obtain the 1-Laplacian in the limit instead of a multiple of it.

Proposition 3.5. Let Ω be a smooth bounded domain in ℝ^{N} and ψ̃ ∈ L^{∞}(∂Ω). Let ψ ∈ W^{1,1}(Ω_{J} \ Ω) ∩ L^{∞}(Ω_{J} \ Ω) such that ψ|_{∂Ω} = ψ̃. Assume J(x) ≥ J(y) if |x| ≤ |y|. Then, for any ϕ ∈ L^{∞}(Ω),

(3.14)  (I + B_{1,ψ}^{J_{1,ε}})^{−1} ϕ → (I + Aψ̃)^{−1} ϕ strongly in L^{1}(Ω) as ε → 0.

Proof. Given ε > 0, we set u_{ε} = (I + B_{1,ψ}^{J_{1,ε}})^{−1} ϕ and denote

Ω_{ε} := Ω_{J_{1,ε}} = Ω + supp(J_{1,ε}).

Then, there exists g_{ϵ} ∈ L^{∞}(Ω_{ε} × Ω_{ε}), g_{ε}(x, y) = −g_{ε}(y, x) for almost all (x, y) ∈ Ω_{ε} × Ω_{ε}, ∥g_{ε}∥_{∞} ≤ 1, such that

J (\frac{x − y}{ε} ) g_{ε}(x, y) ∈ J (\frac{x − y}{ε} ) sign(u_{ε}(y) − u_{ε}(x)) a.e. (x, y) ∈ Ω × Ω,
J (\frac{x − y}{ε} ) g_{ε}(x, y) ∈ J (\frac{x − y}{ε} ) sign(ψ̃(y) − u_{ε}(x))  a.e. (x, y) ∈ Ω × (Ω_{ϵ} \ Ω)

and

(3.15)  u_{ε}(x) − \frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}}J (\frac{x − y}{ε} ) g_{ε}(x, y) dy = ϕ(x) a.e. x ∈ Ω.

Therefore, for v ∈ L^{∞}(Ω_{J}), we can write

(3.16) ∫_{Ω} u_{ε}(x)v(x) dx − \frac{C_{J,1}}{ε^{1+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)v(x) dy dx = ∫_{Ω} ϕ(x)v(x) dx.

Observe that we can extend g_{ε} to a function in L^{∞}(Ω_{J} × Ω_{J}), g_{ε}(x, y) = −g_{ε}(y, x) for almost all (x, y) ∈ Ω_{J} × Ω_{J}, ∥g_{ε}∥_{L^{∞}(Ω_{J})} ≤ 1, such that

J (\frac{x − y}{ε} ) g_{ε}(x, y) ∈ J (\frac{x − y}{ε} ) sign((u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)) a.e. (x, y) ∈ Ω_{J} × Ω_{J}.

Let M := max{∥ϕ∥_{L^{∞}(Ω)}, ∥ψ∥_{L^{∞}(Ω_{J})}}. Taking v = (u_{ε} − M)^{+} in (3.16), we get

\begin{matrix}∫_{Ω} u_{ε}(x)(u_{ε}(x) − M)^{+}dx − \frac{C_{J,1}}{ε^{1+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)(u_{ε}(x) − M)^{+} dy dx \\ = ∫_{Ω} ϕ(x)(u_{ε}(x) − M)^{+} dx. \end{matrix}

Now

\begin{matrix}−\frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)((u_{ε})_{ψ}(x) − M)^{+} dy dx \\ = \frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)(((u_{ε})_{ψ}(y) − M)^{+} − ((u_{ε})_{ψ}(x) − M)^{+}) dy dx ≥ 0. \end{matrix}

Hence, we get

∫_{Ω} u_{ε}(x)(u_{ε}(x) − M)^{+}dx ≤ ∫_{Ω} ϕ(x)(u_{ε}(x) − M)^{+}dx.

Consequently

0 ≤ ∫_{Ω}(u_{ε}(x) − M)(u_{ε}(x) − M)^{+}dx ≤ ∫_{Ω}(ϕ(x) − M)(u_{ε}(x) − M)^{+}dx ≤ 0,

and we deduce u_{ε}(x) ≤ M for almost all x ∈ Ω. Analogously, we can obtain −M ≤ u_{ε}(x) for almost all x ∈ Ω. Thus

(3.17)  ∥u_{ε}∥_{L^{∞}(Ω)} ≤ M for all ϵ > 0,

from here, we can assume there exists a sequence ε_{n} → 0 such that

u_{ε_{n}} ⇀ u weakly in L^{1}(Ω).

Taking v = u_{ε} in (3.16), we have

(3.18) ∫_{Ω} u_{ε}(x)u_{ε}(x) dx − \frac{C_{J,1}}{ε^{1+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y) dyu_{ε}(x) dx = ∫_{Ω} ϕ(x)u_{ε}(x)dx.

Observe that,

−\frac{C_{J,1}}{ε^{1+N}} _{∫Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)dyu_{ε}(x) dx

= −\frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)dy(u_{ε})_{ψ}(x) dx

+\frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}\Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)dyψ(x) dx.

20 F. ANDREU, J. M. MAZÓN, J. D. ROSSI AND J. TOLEDO

Then,

|\frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}\Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)dyψ(x)dx|

≤ \frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}\Ω} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) dy|ψ(x)|dx

≤ \frac{C_{J,1}}{ε} M ∫_{Ω_{ε}\Ω} (\frac{1}{ε^{N}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) dy) dx ≤ \frac{C_{J,1}}{ε} M|Ω_{ε} \ Ω| ≤ M_{1}.

On the other hand,

\begin{matrix}−\frac{C_{J,1}}{ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) g_{ε}(x, y)dy(u_{ε})_{ψ}(x)dx \\ = \frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx. \end{matrix}

Consequently, from (3.17) and (3.18), it follows that

(3.19)  \frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx ≤ M_{2}.

Let us compute,

\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J}} ∫_{Ω_{J}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

= \frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

+\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

+\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

+\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx.

Now, since ψ ∈ W^{1,1}(Ω_{J} \ Ω), we get

\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

= \frac{C_{J,1}}{2ε^{N}}_{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} J (\frac{x − y}{ε} ) \frac{|ψ(y) − ψ(x)|}{ε}dydx ≤ M_{3}.

On the other hand, we have

\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx

= \frac{C_{J,1}}{2ε^{N}}_{∫Ω_{J} \Ω_{ε}} ∫_{Ω_{ε}\Ω} J (\frac{x − y}{ε} ) \frac{|ψ(y) − ψ(x)|}{ε} dy dx

≤ M_{4} \frac{C_{J,1}}{2} ∫_{Ω_{J} \Ω_{ε}} (\frac{1}{ε^{N}} ∫_{Ω_{ε}} J (\frac{x − y}{ε} ) dy) dx ≤ M_{5}.

With similar arguments we obtain,

\frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{ε}} ∫_{Ω_{J} \Ω_{ε}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx ≤ M_{6}.

Therefore,

(3.20)  \frac{C_{J,1}}{2ε^{1+N}} _{∫Ω_{J}} ∫_{Ω_{J}} J (\frac{x − y}{ε} ) |(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)| dy dx ≤ M_{7}.

In particular, we get

∫_{Ω_{J}} ∫_{Ω_{J}} \frac{1}{2} \frac{C_{J,1}}{ε^{N}} J (\frac{x − y}{ε} ) |\frac{(u_{ε})_{ψ}(y) − (u_{ε})_{ψ}(x)}{ε}| dx dy ≤ M_{7} ∀ n ∈ ℕ.

By Proposition 2.8, there exists a subsequence, denote equal, and w ∈ BV (Ω_{J}) such that

(u_{ε_{n}})_{ψ} → w strongly in L^{1}(Ω_{J})

and

(3.21)  \frac{C_{J,1}}{2} J(z)χ_{Ω}(x + ε_{n}z)\frac{u_{ε_{n}}(x + ε_{n}z) − u_{ε_{n}}(x)}{ε_{n}} ⇀ \frac{C_{J,1}}{2} J(z)z · Dw

weakly as measures. Hence, it is easy to obtain that

w(x) = u_{ψ}(x) = \{ \begin{matrix}u(x) & if x ∈ Ω, \\ ψ(x) & if x ∈ Ω_{J} \ Ω, \end{matrix}

and u ∈ BV (Ω).

Moreover, we can also assume that

(3.22)  J(z)χ_{Ω_{J}} (x + ε_{n}z)g_{ε_{n}}(x, x + ε_{n}z) ⇀ Λ(x, z)

weakly^{∗} in L^{∞}(Ω_{J}) × L^{∞}(ℝ^{N}), and Λ(x, z) ≤ J(z) almost every where in Ω_{J} × ℝ^{N}. Taking in (3.16) v ∈ D(Ω), we get for ε = ε_{n} small enough

(3.23) ∫_{Ω} u_{ε_{n}}(x)v(x)dx − \frac{C_{J,1}}{ε_{n}^{1+N}} _{∫Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)v(x) dy dx = ∫_{Ω} ϕ(x)v(x) dx.

Changing variables and taking into account (3.23), we can write

(3.24)  \begin{matrix}\frac{C_{J,1}}{2}_{∫ℝ^{N}} _{∫Ω}J(z)χ_{Ω}(x + ε_{n}z)g_{ε_{n}}(x, x + ε_{n}z) dz \frac{v(x + ε_{n}z) − v(x)}{ε_{n}} dx \\ = −\frac{C_{J,1}}{ε_{n}}_{∫ℝ^{N}} _{∫Ω}J(z)χ_{Ω}(x + ε_{n}z)g_{ε_{n}}(x, x + ε_{n}z) dz v(x) dx \\ = ∫_{Ω}(ϕ(x) − u_{ε_{n}}(x))v(x) dx. \end{matrix}

By (3.22), passing to the limit in (3.24), we get

(3.25)  \frac{C_{J,1}}{2}_{∫ℝ^{N}} _{∫Ω}Λ(x, z)z · ∇v(x) dx dz = ∫_{Ω}(ϕ(x) − u(x))v(x) dx ∀v ∈ D(Ω).

We set ζ = (ζ_{1}, …, ζ_{N}), the vector field defined by

ζ_{i}(x) := \frac{C_{J,1}}{2}_{∫ℝ^{N}}Λ(x, z)z_{i} dz, i = 1, …, N.

Then, ζ ∈ L^{∞}(Ω_{J}, ℝ^{N}), and from (3.25),

−div(ζ) = ϕ − u in D′(Ω).

Let us see that

∥ζ∥_{L^{∞}(Ω_{J})} ≤ 1.

22 F. ANDREU, J. M. MAZÓN, J. D. ROSSI AND J. TOLEDO

Given ξ ∈ ℝ^{N} \ {0}, let R_{ξ} be the rotation such that R_{ξ}^{t}(ξ) = e_{1}|ξ|. If we make the change of variables z = R_{ξ}(y), we obtain

ζ(x) · ξ = \frac{C_{J,1}}{2}_{∫ℝ^{N}}Λ(x, z)z · ξ dz = \frac{C_{J,1}}{2}_{∫ℝ^{N}}Λ(x, R_{ξ}(y))R_{ξ}(y) · ξ dy

= \frac{C_{J,1}}{2}_{∫ℝ^{N}}Λ(x, R_{ξ}(y))y_{1}|ξ| dy.

On the other hand, since J is a radial function and Λ(x, z) ≤ J(z) almost every where,

C_{J,1}^{−1} = \frac{1}{2} ∫_{ℝ^{N}} J(z)|z_{1}| dz

and

|ζ(x) · ξ| ≤ \frac{C_{J,1}}{2}_{∫ℝ^{N}}J(y)|y_{1}| dy|ξ| = |ξ| a.e. x ∈ Ω_{J}.

Therefore, ∥ζ∥_{L^{∞}(Ω_{J})} ≤ 1.

To finish the proof, that is to show that u = (I + Aψ̃)^{−1} ϕ, since u ∈ L^{∞}(Ω) and ψ̃ ∈ L^{∞}(∂Ω), we only need to prove that

(3.26)  (ζ, Du) = |Du| as measures in Ω

and

(3.27)  [ζ, ν] ∈ sign(ψ̃ − u) H^{N−1} − a.e. on ∂Ω.

Given 0 ≤ φ ∈ D(Ω), taking ε = ε_{n} and v = φu_{ε_{n}} in (3.16), we get

(3.28)  \begin{matrix}−\frac{C_{J,1}}{ε_{n}^{1+N}} ∫_{Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)u_{ε_{n}}(x)φ(x) dy dx \\ = \frac{C_{J,1}}{2ε_{n}^{1+N}} ∫_{Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)(u_{ε_{n}}(y)φ(y) − u_{ε_{n}}(x)φ(x)) dy dx \\ = ∫_{Ω}(ϕ(x) − u_{ε_{n}}(x))u_{ε_{n}}(x)φ(x) dx. \end{matrix}

Now, we decompose the double integral as follows,

I_{n} := \frac{C_{J,1}}{2ε_{n}^{1+N}} ∫_{Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)(u_{ε_{n}}(y)φ(y) − u_{ε_{n}}(x)φ(x)) dy dx = I_{n}^{1} + I_{n}^{2},

where

\begin{matrix}I_{n}^{1} := & \frac{C_{J,1}}{2ε_{n}^{1+N}} ∫_{Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) |u_{ε_{n}}(y) − u_{ε_{n}}(x)|φ(y) dy dx \\ = & \frac{C_{J,1}}{2} ∫_{Ω} ∫_{Ω} J (z) χ_{Ω}(x + ε_{n}z)\frac{|u_{ε_{n}}(x + ε_{n}z) − u_{ε_{n}}(x)|}{ε_{n}}φ(x + ε_{n}z) dz dx \end{matrix}

and

\begin{matrix}I_{n}^{2} := & \frac{C_{J,1}}{2ε_{n}^{1+N}} ∫_{Ω} ∫_{Ω} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)u_{ε_{n}}(x)(φ(y) − φ(x)) dy dx \\ = & \frac{C_{J,1}}{2} ∫_{Ω} ∫_{Ω} J(z)χ_{Ω}(x + ε_{n}z)g_{ε_{n}}(x, x + ε_{n}z)u_{ε_{n}}(x)\frac{φ(x + ε_{n}z) − φ(x)}{ε_{n}} dz dx. \end{matrix}

Having in mind (3.21), it follows that

lim_{n→∞} I_{n}^{1} ≥ \frac{C_{J,1}}{2} ∫_{Ω} ∫_{Ω} J (z) φ(x)|z · Du| = ∫_{Ω} φ |Du|.

On the other hand, since

u_{ε_{n}} → u strongly in L^{1}(Ω),

by (3.22), we get

lim_{n→∞} I_{n}^{2} = \frac{C_{J,1}}{2} ∫_{Ω} ∫_{ℝ^{N}} u(x)Λ(x, z)z · ∇φ(x) dz dx = ∫_{Ω} u(x)ζ(x) · ∇φ(x) dx.

Therefore, taking n → +∞ in (3.28), we obtain

(3.29)  ∫_{Ω} φ |Du| + ∫_{Ω} u(x)ζ(x) · ∇φ(x) dx ≤ ∫_{Ω}(ϕ(x) − u(x))u(x)φ(x) dx.

By Green’s formula,

∫_{Ω}(ϕ(x) − u(x))u(x)φ(x)dx = − ∫_{Ω} div(ζ)uφ dx = ∫_{Ω}(ζ, D(φu))
= ∫_{Ω} φ(ζ, Du) + ∫_{Ω} u(x)ζ(x) · ∇φ(x) dx.

Since |(ζ, Du)| ≤ |Du|, the last identity and (3.29) give (3.26).

Finally, we show that (3.27) holds. We take w_{m} ∈ W^{1,1}(Ω) ∩ C(Ω) such that w_{m} = ψ̃ H^{N−1}-a.e. on ∂Ω, and w_{m} → u in L^{1}(Ω). Taking v = v_{m,n} := (u_{ε_{n}})_{ψ} − (w_{m})_{ψ} in (3.16), we get

(3.30) \begin{matrix}∫_{Ω}(ϕ(x) − u_{ε_{n}}(x))(u_{ε_{n}}(x) − w_{m}(x)) dx \\ = −\frac{C_{J,1}}{ε_{n}^{1+N}} _{∫Ω_{J}} ∫_{Ω_{J}} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)v_{m,n}(x) dy dx \\ = \frac{C_{J,1}}{2ε_{n}^{1+N}} _{∫Ω_{J}} ∫_{Ω_{J}} J (\frac{x − y}{ε_{n}} ) g_{ε_{n}}(x, y)(v_{m,n}(x) − v_{m,n}(x)) dy dx = H_{n}^{1} + H_{m,n}^{1}, \end{matrix}

where

H_{n}^{1} = \frac{C_{J,1}}{2}_{∫Ω_{J}} _{∫ℝ^{N}}J(z)χ_{Ω_{J}} (x + ε_{n}z) |\frac{(u_{ε_{n}})_{ψ}(x + ε_{n}z) − (u_{ε_{n}})_{ψ}(x)}{ε_{n}} | dz dx

and

H_{m,n}^{2} = −\frac{C_{J,1}}{2}_{∫Ω_{J}} _{∫ℝ^{N}}J(z)χ_{Ω_{J}} (x + ε_{n}z)g_{ε_{n}}(x, x + ε_{n}z)\frac{(w_{m})_{ψ}(x + ε_{n}z) − (w_{m})_{ψ}(x)}{ε_{n}} dz dx.

Arguing as before,

lim_{n→∞} H_{n}^{1} ≥ _{∫Ω_{J}} |Du_{ψ}| = _{∫Ω}|Du| + ∫_{∂Ω} |u − ψ̃| dH^{N−1} + ∫_{Ω_{J} \Ω} |∇ψ|.

On the other hand, since (w_{m})_{ψ} ∈ W^{1,1}(Ω_{J}), by (3.22),

lim_{n→∞} H_{m,n}^{2} = −\frac{C_{J,1}}{2}_{∫Ω_{J}} _{∫ℝ^{N}}Λ(x, z)z · ∇(w_{m})_{ψ}(x) dz dx = −_{∫Ω_{J}} ζ(x) · ∇(w_{m})_{ψ}(x) dx.

Consequently, taking n → ∞ in (3.30), we get

(3.31)  \begin{matrix}∫_{Ω}(ϕ(x) − u(x))(u(x) − w_{m}(x)) dx \\ ≥ _{∫Ω}|Du| +_{∫∂Ω}|u − ψ̃| dH^{N−1} + ∫_{Ω_{J} \Ω} |∇ψ| −_{∫Ω_{J}} ζ(x) · ∇(w_{m})_{ψ}(x) dx. \end{matrix}

Now,

−_{∫Ω_{J}} ζ(x) · ∇(w_{m})_{ψ}(x) dx = −_{∫Ω}ζ(x) · ∇w_{m}(x) dx −_{∫Ω_{J} \Ω}ζ(x) · ∇ψ(x) dx
= _{∫Ω}divζ(x)w_{m}(x) dx − ∫_{∂Ω}[ζ, ν]ψ̃ dH^{N−1} −_{∫Ω_{J} \Ω}ζ(x) · ∇ψ(x) dx.

Since

∫_{Ω_{J} \Ω} |∇ψ| −_{∫Ω_{J} \Ω}ζ(x) · ∇ψ(x) dx ≥ 0,

from (3.31), we have

∫_{Ω}(ϕ(x) − u(x))(u(x) − w_{m}(x)) dx
≥ _{∫Ω}|Du| + ∫_{∂Ω} |u − ψ̃| dH^{N−1} +_{∫Ω}divζ(x)w_{m}(x) dx − ∫_{∂Ω}[ζ, ν]ψ̃ dH^{N−1}.

Letting m → ∞, and using Green’s formula, we deduce

0 ≥ _{∫Ω}|Du| + ∫_{∂Ω} |u − ψ̃| dH^{N−1} +_{∫Ω}divζ(x)u(x) dx − ∫_{∂Ω}[ζ, ν]ψ̃ dH^{N−1}

= _{∫Ω}|Du| + ∫_{∂Ω} |u − ψ̃| dH^{N−1} − ∫_{Ω}(ζ, Du) + ∫_{∂Ω}[ζ, ν]u dH^{N−1} − ∫_{∂Ω}[ζ, ν]ψ̃ dH^{N−1}.

By (3.26), we obtain

∫_{∂Ω} |u − ψ̃| dH^{N−1} ≤ ∫_{∂Ω}[ζ, ν](ψ̃ − u) dH^{N−1} ≤ ∫_{∂Ω} |u − ψ̃| dH^{N−1}.

Therefore,

[ζ, ν] ∈ sign(ψ̃ − u) H^{N−1} − a.e. on ∂Ω,

and the proof is finished. □

From the above Proposition, by standard results of the Nonlinear Semigroup Theory (see,

[20] or [17]), we obtain Theorem 1.6.

4. Limit as p → +∞. A Model for Sandpiles

4.1. A model for sandpiles. Let sand be poured out onto a rigid surface, y = u_{0}(x), given in a bounded open subset Ω of ℝ^{2} with Lipschitz boundary ∂Ω. If the support boundary is open and we assume that the angle of stability is equal to \frac{π}{4}, a model for pile surface evolution was proposed by L. Prigozhin [36] as

(4.1)  ∂_{t}u + div q = f, u|_{t=0} = u_{0}, u|_{∂Ω} = u_{0}|_{∂Ω},

where u(t, x) is the unknown pile surface, f(t, x) ≥ 0 is the given source density and q(t, x) is the unknown horizontal projection of the flux of sand pouring down the pile surface. If the support has slope steeper than the sand angle of repose, ∥∇u_{0}∥_{∞} ≤ 1, Prigozhin ([36], see also [12], [30] and the references therein) proposed to take q = −m∇u, where m is the Lagrange multiplier related to the constraint ∥∇u_{0}∥_{∞} ≤ 1 and satisfies m(∥∇u∥^{2} − 1) = 0 and reformulated this model as the following variational inequality:

(4.2)  \{ \begin{matrix}f(t, ·) − u_{t}(t) ∈ ∂I_{K(u_{0})}(u(t)), a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x), \end{matrix}

where

K(u_{0}) := {v ∈ W^{1,∞}(Ω) : ∥∇v∥_{∞} ≤ 1, v|_{∂Ω} = u_{0}|_{∂Ω}} .

Our aim is to approximate the Prigozhin model for the sandpile by a nonlocal model (Theorem 1.8) obtained as the limit as p → +∞ of the nonlocal p-Laplacian problem with Dirichlet boundary condition (Theorem 1.7).

To identify the limit as p → +∞ of the solutions u_{p} of problem P_{p}^{J}(u_{0}, ψ) we will use the methods of convex analysis, and so we first recall some terminology (see [31], [19] and [10]). If H is a real Hilbert space with inner product ( , ) and Ψ : H → (−∞, +∞] is convex, then the subdifferential of Ψ is defined as the multivalued operator ∂Ψ given by

v ∈ ∂Ψ(u) ⟺ Ψ(w) − Ψ(u) ≥ (v, w − u) ∀ w ∈ H.

The epigraph of Ψ is defined by Epi(Ψ) = {(u, λ) ∈ H × ℝ : λ ≥ Ψ(u)}.

Given K a closed convex subset of H, the indicator function of K is defined by

I_{K}(u) = \{ \begin{matrix}0 & if u ∈ K, \\ +∞ & if u ∉ K. \end{matrix}

Then it is easy to see that the subdifferential is characterized as follows,

v ∈ ∂I_{K}(u) ⟺ u ∈ K and (v, w − u) ≤ 0 ∀ w ∈ K.

In case the convex functional Ψ : H → (−∞, +∞] is proper , lower-semicontinuous and min Ψ = 0 , it is well known (see [19]) that the abstract Cauchy problem

\{ \begin{matrix}u′(t) + ∂Ψ(u(t)) ∋ f(t), & a.e t ∈]0, T [, \\ u(0) = u_{0}, \end{matrix}

has a unique strong solution for any f ∈ L^{2}(0, T; H) and u_{0} ∈ \overline{D(∂Ψ)}.

The following convergence was studied by Mosco in [35] (see [10]). Suppose X is a metric space and A_{n} ⊂ X. We define

lim inf_{n→∞} A_{n} = {x ∈ X : ∃x_{n} ∈ A_{n}, x_{n} → x}

and

lim sup_{n→∞} A_{n} = {x ∈ X : ∃x_{n_{k}} ∈ A_{n_{k}}, x_{n_{k}} → x}.

In the case X is a normed space, we note by s − lim and w − lim the above limits associated respectively to the strong and to the weak topology of X.

Given a sequence Ψ_{n}, Ψ : H → (−∞, +∞] of convex lower-semicontinuous functionals, we say that Ψ_{n} converges to Ψ in the sense of Mosco if

(4.3)  w − lim sup_{n→∞} Epi(Ψ_{n}) ⊂ Epi(Ψ) ⊂ s − lim inf_{n→∞} Epi(Ψ_{n}).

It is easy to see that (4.3) is equivalent to the two following conditions:

(4.4)  ∀ u ∈ D(Ψ) ∃u_{n} ∈ D(Ψ_{n}) : u_{n} → u and Ψ(u) ≥ lim sup_{n→∞} Ψ_{n}(u_{n});
(4.5)  for every subsequence n_{k}, when u_{k} ⇀ u, it holds Ψ(u) ≤ lim inf_{k} Ψ_{n_{k}}(u_{k}).

As consequence of results in [20] and [10] we can write the following result.

Theorem 4.1. Let Ψ_{n}, Ψ : H → (−∞, +∞] convex lower-semicontinuous functionals. Then the following statements are equivalent

(i) Ψ_{n} converges to Ψ in the sense of Mosco.

(ii) (I + λ∂Ψ_{n})^{−1}u → (I + λ∂Ψ)^{−1}u, ∀ λ > 0, u ∈ H.

Moreover, any of these two conditions (i) or (ii) imply that

(iii) for every u_{0} ∈ \overline{D(∂Ψ)} and u_{0,n} ∈ \overline{D(∂Ψ_{n})} such that u_{0,n} → u_{0}, and every f_{n}, f ∈ L^{2}(0, T; H) with f_{n} → f, if u_{n}(t), u(t) are the strong solutions of the abstract Cauchy problems

\{ \begin{matrix}u′_{n}(t) + ∂Ψ_{n}(u_{n}(t)) ∋ f_{n}, & a.e. t ∈]0, T [, \\ u_{n}(0) = u_{0,n}, \end{matrix}

and

\{ \begin{matrix}u′(t) + ∂Ψ(u(t)) ∋ f, & a.e. t ∈]0, T [, \\ u(0) = u_{0}, \end{matrix}

respectively, then

u_{n} → u in C([0, T] : H).

4.2. Limit as p → +∞. Let us consider the nonlocal p-Laplacian evolution problem with source

P_{p}^{J}(u_{0}, ψ, f) \begin{cases}u_{t}(t, x)=∫_{Ω}J(x − y)|u(t, y) − u(t, x)|^{p−2}(u(t, y) − u(t, x))dy + f(t, x), \\ (t, x) ∈]0, T [×Ω, \\ u(t, x) = ψ(x), (t, x) ∈]0, T [×(Ω_{J} \ \overline{Ω}), \\ u(0, x) = u_{0}(x), x ∈ Ω. \end{cases}

This problem is associated to the energy functional

G_{p,ψ}^{J}(u) = \frac{1}{2p} ∫_{Ω} ∫_{Ω} J(x − y)|u(y) − u(x)|^{p} dy dx + \frac{1}{p} ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y)|ψ(y) − u(x)|^{p} dy dx.

With a formal calculation, taking limit as p → +∞, we arrive to the functional

G_{∞,ψ}^{J}(u) = \{ \begin{matrix}0 & if |u(x) − u(y)| ≤ 1, for x, y ∈ \overline{Ω} \\ & and |ψ(y) − u(x)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ & with x − y ∈ supp(J) \\ +∞ & in other case. \end{matrix}

Hence, if we define

K_{∞,ψ}^{J} := \{u ∈ L^{2}(Ω) : \begin{matrix}|u(x) − u(y)| ≤ 1, x, y ∈ Ω \\ and |ψ(y) − u(x)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ with x − y ∈ supp(J) \end{matrix} \} ,

we have that the functional G_{∞,ψ}^{J} is given by the indicator function of K_{∞,ψ}^{J}, that is, G_{∞,ψ}^{J} = I_{K_{∞,ψ}^{J}}. Then, the nonlocal limit problem can be written as

P_{∞}^{J}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{t}(t) ∈ ∂I_{K_{∞,ψ}^{J}}(u(t)), a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x). \end{matrix}

Proof of Theorem 1.7. Let T > 0. By Theorem 4.1, to prove the result it is enough to show that the functionals

G_{p,ψ}^{J}(u) = \frac{1}{2p} ∫_{Ω} ∫_{Ω} J(x − y)|u(y) − u(x)|^{p} dy dx + \frac{1}{p} ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y)|ψ(y) − u(x)|^{p} dy dx

converge to

G_{∞,ψ}^{J}(u) = \{ \begin{matrix}0 & if |u(x) − u(y)| ≤ 1, for x, y ∈ Ω \\ & and |ψ(y) − u(x)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ Ω, \\ & with x − y ∈ supp(J) \\ +∞ & in other case. \end{matrix}

as p → +∞, in the sense of Mosco. First, let us check that

(4.6)  Epi(G_{∞,ψ}^{J}) ⊂ s − lim inf_{p→+∞} Epi(G_{p,ψ}^{J}).

To this end let (u, λ) ∈ Epi(G_{∞,ψ}^{J}). We can assume that u ∈ K_{∞,ψ}^{J} and λ ≥ 0 (as G_{∞,ψ}^{J}(u) = 0). Now take

(4.7)  v_{p} = u and λ_{p} = G_{p,ψ}^{J}(u) + λ.

Then, as λ ≥ 0 we have (v_{p}, λ_{p}) ∈ Epi(G_{p,ψ}^{J}). Obviously, v_{p} = u → u in L^{2}(Ω), and as u ∈ K_{∞,ψ}^{J},

G_{p,ψ}^{J}(u) = \frac{1}{2p} ∫_{Ω} ∫_{Ω} J(x − y)|u(y) − u(x)|^{p} dydx

+\frac{1}{p} ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y)|ψ(y) − u(x)|^{p} dy dx

≤ \frac{1}{2p} ∫_{Ω} ∫_{Ω} J(x − y) dydx + \frac{1}{p} ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y) dy dx → 0

as p → +∞. Therefore, we get (4.6). Finally, let us prove that

(4.8)  w − lim sup_{p→+∞} Epi(G_{p,ψ}^{J}) ⊂ Epi(G_{∞,ψ}^{J}).

To this end, let us consider a sequence (u_{p_{j}}, λ_{p_{j}}) ∈ Epi(G_{p_{j},ψ}^{J}), that is, G_{p_{j},ψ}^{J}(u_{p_{j}}) ≤ λ_{p_{j}}, with

u_{p_{j}} ⇀ u, and λ_{p_{j}} → λ.

Since, 0 ≤ G_{p_{j},ψ}^{J}(u_{p_{j}}) ≤ λ_{p_{j}} → λ, 0 ≤ λ. On the other hand, we have that there exists a constant C > 0 such that

\begin{matrix}(p_{j}C)^{1/p_{j}} ≥ (p_{j}G_{p,ψ}^{J}(u_{p_{j}}))^{1/p_{j}} = & (\frac{1}{2} ∫_{Ω} ∫_{Ω} J(x − y)|u_{p_{j}}(y) − u_{p_{j}}(x)|^{p_{j}} dy dx \\ + ∫_{Ω} ∫_{Ω_{J} \Ω} J(x − y)|ψ(y) − u_{p_{j}}(x)|^{p_{j}} dy dx) & 1/p_{j} \end{matrix}

Then, by the above inequality,

(∫_{Ω} ∫_{Ω} J (x − y) |u_{p_{j}}(y) − u_{p_{j}}(x)|^{q} dy dx)^{1/q} ≤ (∫_{Ω} ∫_{Ω} J (x − y) dy dx)^{(p_{j}−q)/p_{j}q}

× (∫_{Ω} ∫_{Ω} J (x − y) |u_{p_{j}}(y) − u_{p_{j}}(x)|^{p_{j}} dy dx)^{1/p_{j}}

≤ (∫_{Ω} ∫_{Ω} J (x − y) dy dx)^{(p_{j}−q)/p_{j}q} (Cp_{j})^{1/p_{j}}.

Hence, we can extract a subsequence (if necessary) and let p_{j} → +∞ to obtain

(∫_{Ω} ∫_{Ω} J (x − y) |u(y) − u(x)|^{q} dy dx)^{1/q} ≤ (∫_{Ω} ∫_{Ω} J (x − y) dy dx)^{1/q} .

Now, just taking q → +∞, we get

|u(x) − u(y)| ≤ 1 a.e. (x, y) ∈ Ω × Ω, x − y ∈ supp(J).

With a similar argument we obtain

|u(x) − ψ(y)| ≤ 1 a.e. x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, with x − y ∈ supp(J).

Hence, we conclude that u ∈ K_{∞,ψ}^{J}. This ends the proof. □

4.3. Rescaling. We will assume now that Ω is convex and ψ verifies ∥∇ψ∥_{∞} ≤ 1. For ε > 0, we rescale the functional G_{∞,ψ}^{J} as follows

G_{∞,ψ}^{ε}(u) = \{ \begin{matrix}0 & if |u(x) − u(y)| ≤ ε, for x, y ∈ Ω \\ & and |ψ(y) − u(x)| ≤ ε, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ & with |x − y| ≤ ε \\ +∞ & in other case. \end{matrix}

In other words, G_{∞,ψ}^{ε} = I_{K_{∞,ψ}^{ε}}, where

K_{∞,ψ}^{ε} := \{u ∈ L^{2}(Ω) : \begin{matrix}|u(x) − u(y)| ≤ ε, x, y ∈ Ω \\ and |ψ(y) − u(x)| ≤ ε, for x ∈ Ω, y ∈ Ω_{J} \ \overline{Ω}, \\ with |x − y| ≤ ε \end{matrix} \} .

Consider the gradient flow associated to the functional G_{∞,ψ}^{ε}

P_{∞}^{ε}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{t}(t, ·) ∈ ∂I_{K_{∞,ψ}^{ε}}(u(t)), & a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x), & in Ω, \end{matrix}

and the problem

P_{∞}(u_{0}, ψ, f) \{ \begin{matrix}f(t, ·) − u_{∞,t} ∈ ∂I_{K_{ψ}}(u_{∞}), & a.e. t ∈]0, T [, \\ u_{∞}(0, x) = u_{0}(x), & in Ω, \end{matrix}

where

K_{ψ} := {u ∈ W^{1,∞}(Ω) : ∥∇u∥_{∞} ≤ 1, u|_{∂Ω} = ψ|_{∂Ω}} .

Observe that if u ∈ K_{ψ}, ∥∇u∥_{∞} ≤ 1. Then, since ∥∇ψ∥_{∞} ≤ 1 and Ω is convex, we have |u(x) − u(y)| ≤ |x − y| and |u(x) − ψ(y)| ≤ |x − y|, from where it follows that u ∈ K_{∞,ψ}^{ε}, that is, K_{ψ} ⊂ K_{∞,ψ}^{ε}.

With all these definitions and notations, we can proceed with the limit as ε → 0 for the sandpile model (p = +∞).

Proof of Theorem 1.8. Since u_{0} ∈ K_{ψ}, u_{0} ∈ K_{∞,ψ}^{ε} for all ε > 0. Again we are using that ∥∇ψ∥_{∞} ≤ 1. Consequently there exists u_{∞,ε} the unique solution of P_{∞}^{ε}(u_{0}, ψ, f).

By Theorem 4.1, to prove the result it is enough to show that I_{K_{∞,ψ}^{ε}} converges to I_{K_{ψ}} in the sense of Mosco. Using that ∥∇ψ∥_{∞} ≤ 1 it is easy to obtain that

(4.9)  K_{∞,ψ}^{ε_{1}} ⊂ K_{∞,ψ}^{ε_{2}}, if ε_{1} ≤ ε_{2}.

Since K_{ψ} ⊂ K_{∞,ψ}^{ε} for all ε > 0, we have

K_{ψ} ⊂ ⋂_{ε>0} K_{∞,ψ}^{ε}.

On the other hand, if

u ∈ ⋂_{ε>0} K_{∞,ψ}^{ε},

we have

|u(y) − u(x)| ≤ |y − x|, a.e x, y ∈ Ω,

and moreover

|u(y) − ψ(x)| ≤ |y − x|, a.e x ∈ Ω_{J} \ Ω, y ∈ Ω,

from where it follows that u ∈ K_{ψ}. Therefore, we have

(4.10)  K_{ψ} = ⋂_{ε>0} K_{∞,ψ}^{ε}.

Note that

(4.11)  Epi(I_{K_{ψ}}) = K_{ψ} × [0, ∞[, Epi(I_{K_{∞,ψ}^{ε}}) = K_{∞,ψ}^{ε} × [0, ∞[ ∀ ε > 0.

By (4.10) and (4.11),

(4.12)  Epi(I_{K_{ψ}}) ⊂ s − lim inf_{ε→0} Epi(I_{K_{∞,ψ}^{ε}}).

On the other hand, given (u, λ) ∈ w − lim sup_{ε→0} Epi(I_{K_{∞,ψ}^{ε}}) there exists (u_{ε_{k}}, λ_{k}) ∈ K_{ε_{k},ψ} × [0, ∞[, such that ε_{k} → 0 and

u_{ε_{k}} ⇀ u in L^{2}(Ω), λ_{k} → λ in ℝ.

By (4.9), given ε > 0, there exists k_{0}, such that u_{ε_{k}} ∈ K_{∞,ψ}^{ε} for all k ≥ k_{0}. Then, since K_{∞,ψ}^{ε} is a closed convex set, we get u ∈ K_{∞,ψ}^{ε}, and, by (4.10), we obtain that u ∈ K_{0}. Consequently,

(4.13)  w − lim sup_{n→∞} Epi(I_{K_{∞,ψ}^{ε}}) ⊂ Epi(I_{K_{ψ}}).

Finally, by (4.12), (4.13) and having in mind (4.3), we obtain that I_{K_{∞,ψ}^{ε}} converges to I_{K_{ψ}} in the sense of Mosco. □

4.4. Explicit solutions. Our goal now is to show some explicit examples that illustrate the behavior of the solutions when p = +∞.

Remark 4.2. There is a natural upper bound (and of course also a natural lower bound) for the solutions with boundary datum ψ outside Ω (regardless the source term f). Indeed, given a bounded domain Ω ⊂ ℝ^{N} let us define inductively

Ω_{1} = \{x ∈ Ω : |x − y| < 1 for some y ∈ Ω_{J} \ \overline{Ω}\}

and, for j ≥ 2,

Ω_{j} = {x ∈ Ω \ ∪_{i=1}^{j−1}Ω_{i} : |x − y| < 1 for some y ∈ Ω_{j−1}} .

Then, since u(t) ∈ K_{∞,ψ}^{J} we must have

u(t, x) ≤ ψ(y) + 1 if |x − y| ≤ 1, x ∈ Ω_{1}, y ∈ Ω_{J} \ \overline{Ω},

and for any j ≥ 2

u(t, x) ≤ u(t, y) + 1 if |x − y| ≤ 1, x ∈ Ω_{j}, y ∈ Ω_{j−1} \ Ω_{j}.

Therefore we have an upper bound for u(t, x) in the whole Ω,

u(t, x) ≤ Ψ_{1}(x),

where Ψ_{1} is defined by the inductive formula,

Ψ_{1}(x) = max \{ψ(y) + 1 : y ∈ Ω_{J} \ \overline{Ω}, |x − y| ≤ 1\} , for x ∈ Ω_{1},

and

Ψ_{1}(x) = max {Ψ_{1}(y) + 1 : y ∈ Ω_{j−1}, |x − y| ≤ 1} , for x ∈ Ω_{j}, if j ≥ 2.

Analogously, we can obtain a lower bound for u(t, x),

u(t, x) ≥ Φ_{1}(x),

where Φ_{1} is defined by the inductive formula,

Φ_{1}(x) = min \{ψ(y) − 1 : y ∈ Ω_{J} \ \overline{Ω}, |x − y| ≤ 1\} , for x ∈ Ω_{1},

and

Φ_{1}(x) = min {Φ_{1}(y) − 1 : y ∈ Ω_{j−1}, |x − y| ≤ 1} , for x ∈ Ω_{j}, if j ≥ 2.

With this remark in mind we show some explicit examples of solutions to

P_{∞}^{J}(u_{0}, ψ, f) \{ \begin{matrix}f(t, x) − u_{t}(t, x) ∈ ∂G_{∞,ψ}^{J}(u(t)), & a.e. t ∈]0, T [, \\ u(0, x) = u_{0}(x), & in Ω, \end{matrix}

where

G_{∞,ψ}^{J}(u) = \{ \begin{matrix}0 & if u ∈ L^{2}(Ω), |u(x) − u(y)| ≤ 1, for x, y ∈ Ω, |x − y| ≤ 1, \\ & and |u(x) − ψ(y)| ≤ 1, for x ∈ Ω, y ∈ Ω_{J} \ Ω, |x − y| ≤ 1, \\ +∞ & in other case. \end{matrix}

In order to verify that a function u(t, x) is a solution to P_{∞}^{J}(u_{0}, ψ, f) we need to check that

(4.14)  G_{∞,ψ}^{J}(v) ≥ G_{∞,ψ}^{J}(u) + ⟨f − u_{t}, v − u⟩,  for all v ∈ L^{2}(Ω).

To this end we can assume that v ∈ K_{∞,ψ}^{J} (otherwise G_{∞,ψ}^{J}(v) = +∞ and then (4.14) becomes trivial). Therefore, we need to check that

(4.15)  u(t, ·) ∈ K_{∞,ψ}^{J}

and, by (4.14), that

(4.16)  ∫_{Ω}(f(t, x) − u_{t}(t, x))(v(x) − u(t, x)) dx ≤ 0

for every v ∈ K_{∞,ψ}^{J}.

Example 1. Let us consider a nonnegative source f and as initial condition the upper bound defined in the previous remark, u_{0}(x) = Ψ_{1}(x). Then the solution to P_{∞}^{J}(u_{0}, ψ, f) is given by

u(t, x) ≡ Ψ_{1}(x)

for every t > 0. Indeed, Ψ_{1}(x) ∈ K_{∞,ψ}^{J} and for every v ∈ K_{∞,ψ}^{J} we have that v(x) ≤ Ψ_{1}(x) and therefore,

∫_{Ω}(f(t, x) − u_{t}(t, x))(v(x) − u(t, x)) dx = ∫_{Ω} f(t, x)(v(x) − Ψ_{1}(x)) dx ≤ 0

as we have to show.

In general, given a nonnegative source f supported in D ⊂ Ω, any initial condition u_{0} ∈ K_{∞,ψ}^{J} that verifies u_{0}(x) = Ψ_{1}(x) in D produces a stationary solution u(t, x) ≡ u_{0}(x).

Analogously, it can be shown that u(t, x) ≡ Φ_{1}(x) when u_{0}(x) = Φ_{1}(x) and f(t, x) ≤ 0.

Example 2. Now, let us assume that we are in an interval Ω = (−L, L), ψ = 0, ε = L/n, n ∈ N, u_{0} = 0 which belongs to K_{ε,0}, and the source f is an approximation of a delta function,

f(t, x) = f_{η}(t, x) = \frac{1}{η}χ_{[− \frac{η}{2} , \frac{η}{2} ]}(x), 0 < η ≤ 2ε.

Using the same ideas of [8], it is easy to verify the following general formula that describes the solution of P_{∞}^{ε}(u_{0}, ψ, f) for every t ≥ 0. For any given integer l ≥ 0 we have

u(t, x) = \{ \begin{matrix}lε + k_{l}(t − t_{l}), & x ∈ [−\frac{η}{2}, \frac{η}{2}], \\ (l − 1)ε + k_{l}(t − t_{l}), & x ∈ [−\frac{η}{2} − ε, \frac{η}{2} + ε] \ [−\frac{η}{2}, \frac{η}{2}], \\ … \\ k_{l}(t − t_{l}), & x ∈ [−\frac{η}{2} − lε, \frac{η}{2} + lε] \ [−\frac{η}{2} − (l − 1)ε, \frac{η}{2} + (l − 1)ε], \\ 0, & x ∉ [−\frac{η}{2} − lε, \frac{η}{2} + lε], \end{matrix}

for t ∈ [t_{l}, t_{l+1}), where

k_{l} = \frac{1}{2lε + η} and t_{l+1} = t_{l} + \frac{ε}{k_{l}} , t_{0} = 0.

This general formula is valid until the time at which the solution verifies u(t, x) = Ψ_{ε}(x) for x ∈ [−\frac{η}{2}, \frac{η}{2}] (the support of f), that is, until T = t_{l^{∗}+1}, where

l^{∗} is the first l such that lε + k_{l}(t_{l+1} − t_{l}) = Ψ_{ε}(0)

and

Ψ_{ε} is the natural upper bound defined in Remark 4.2

for the corresponding reescaled kernel. Observe that for this l^{∗}, \frac{η}{2} +l^{∗}ε ≤ L. From that time on the solution is stationary, that is, u(t, x) = u(T, x) for all t > T.

From the above formula, taking limits as η → 0, we get that the expected solution to P_{∞}^{ε}(u_{0}, ψ, δ_{0}) is given, for any given integer l ≥ 1, by

(4.17)  u(t, x) = \{ \begin{matrix}(l − 1)ε + k_{l}(t − t_{l}), & x ∈ [−ε, ε], \\ (l − 2)ε + k_{l}(t − t_{l}), & x ∈ [−2ε, 2ε] \ [−ε, ε], \\ … \\ k_{l}(t − t_{l}), & x ∈ [−lε, lε] \ [−(l − 1)ε, (l − 1)ε], \\ 0, & x ∉ [−lε, lε], \end{matrix}

for t ∈ [t_{l}, t_{l+1}), where k_{l} = \frac{1}{2lε}, t_{l+1} = t_{l} + \frac{ε}{k_{l}}, t_{1} = 0, until T = t_{l^{∗}+1}, where

l^{∗} is the first l such that lε + k_{l}(t_{l+1} − t_{l}) = Ψ_{ε}(0).

And from that time on the solution is stationary, that is, u(t, x) = u(T, x) for all t > T.

Remark that, since the space of functions K_{∞,ψ}^{ε} is not contained into C(ℝ), the formulation (4.16) with f = δ_{0} does not make sense. Hence the function u(t, x) described by (4.17) is to be understood as a generalized solution to P_{∞}^{ε}(u_{0}, ψ, δ_{0}) (it is obtained as a limit of solutions to approximating problems).

Note that the function u(T, x) is a “regular and symmetric pyramid” composed by squares of side ε which is one step below the upper profile Ψ_{ε}.

Recovering the sandpile model as ε → 0. Now, to recover the sandpile model, take the limit as ε → 0 in the previous example to get that u(t, x) → v(t, x), where

v(t, x) = (l − |x|)^{+} for t = l^{2},

until the time at which t = L^{2}, and from that time the solution is stationary.

A similar argument shows that, for any a ∈ (0, L), the generalized solution to P_{∞}^{ε}(0, 0, δ_{a}) converges as ε → 0 to v(t, x), where

v(t, x) = (l − |x − a|)^{+} for t = l^{2},

until the time at which t = (L − a)^{2}, and from that time the solution is stationary. These concrete examples illustrates the general convergence result Theorem 1.8.

Acknowledgements. Part of this work was performed during a visit of JDR to Univ. de Valencia. He wants to thank for the warm hospitality and the stimulating working atmosphere found there. FA, JMM and JT are partially supported by the Spanish MEC and FEDER, project MTM2005-00620. JDR is partially supported by ANPCyT PICT 5009, UBA X066, CONICET (Argentina).

References

[1] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón. The Dirichlet problem for the total variation flow. J. Funct. Anal. 180 (2001), 347–403.

[2] F. Andreu, V. Caselles, and J.M. Mazón. Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, vol. 223, 2004. Birkhauser.

[3] F. Andreu, J.M. Mazón and J. Toledo. Stabilization of solutions of the filtration equation with absorption and non-linear flux, NoDEA 2, (1995), 267–289.

[4] F. Andreu, J.M. Mazón, S. Segura and J. Toledo. Quasilinear Elliptic and Parabolic Equations in L^{1} with Nonlinear Boundary Conditions, Adv. in Math. Sci. and Appl. 7, (1997), 183–213.

[5] F. Andreu, J.M. Mazón, S. Segura and J. Toledo. Existence and uniqueness for a degenerate parabolic equation with L^{1}-data. Trans. Amer. Math. Soc., 351(1), (1999), 285–306.

[6] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo. The Neumann problem for nonlocal nonlinear diffusion equations. J. Evol. Equations. 8(1), (2008), 189–215.

[7] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo. A nonlocal p−Laplacian evolution equation with Newmann boundary conditions. To appear in J. Math. Pures Appl.

[8] F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo. The limit as p → ∞ in a nonlocal p−Laplacian evolution equation. A nonlocal approximation of a model for sandpiles. Preprint.

[9] G. Anzellotti. Pairings Between Measures and Bounded Functions and Compensated Compactness, Ann. di Matematica Pura ed Appl. IV (135) (1983), 293-318.

[10] H. Attouch. Familles d’opérateurs maximaux monotones et mesurabilité. Ann. Mat. Pura Appl., 120 (1979), 35–111.

[11] G. Barles and C. Imbert. Second-order elliptic integro-differential equations: Viscosity solutions theory revisited. Preprint.

[12] J. W. Barrett and L. Prigozhin. Dual formulations in critical state problems. Interfaces and Free Boundaries 8 (2006), 349-370.

[13] P. Bates and A. Chmaj. A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal., 150, (1999), 281–305.

[14] P. Bates, P. Fife, X. Ren and X. Wang. Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal., 138, (1997), 105–136.

[15] Ph. Bénilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J.L Vazquez. An L^{1}-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations, Ann. Scuola Normale Superiore di Pisa, IV, Vol. XXII (1995), 241-273.

[16] Ph. Bénilan and M. G. Crandall. Completely accretive operators. In Semigroup theory and evolution equations (Delft, 1989), volume 135 of Lecture Notes in Pure and Appl. Math., pages 41–75. Dekker, New York, 1991.

[17] Ph. Bénilan, M. G. Crandall and A. Pazy. Evolution Equations Governed by Accretive Operators. Book to appear.

[18] H. Brezis. Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier, 18, (1968), 115–175.

[19] H. Brezis. Opérateur Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, 1973.

[20] H. Brezis and A. Pazy. Convergence and approximation of semigroups of nonlinear operators in Banach spaces. J. Funct. Anal., 9 (1972), 63–74.

[21] L. Caffarelli and L. Silvestre. Regularity theory for fully nonlinear integro-differential equations. Preprint.

[22] L. Caffarelli, S. Salsa and L. Silvestre. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. To appear in Inventiones Mathematicae.

[23] C. Carrillo and P. Fife. Spatial effects in discrete generation population models. J. Math. Biol. 50(2), (2005), 161–188.

[24] E. Chasseigne, M. Chaves and J. D. Rossi. Asymptotic behaviour for nonlocal diffusion equations. J. Math. Pures Appl., 86, (2006), 271–291.

[25] X. Chen. Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differential Equations, 2, (1997), 125–160.

[26] C. Cortazar, M. Elgueta and J. D. Rossi. A non-local diffusion equation whose solutions develop a free boundary. Annales Henri Poincaré, 6(2), (2005), 269–281.

[27] C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski. Boundary fluxes for non-local diffusion. J. Differential Equations, 234, (2007), 360–390.

[28] C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rat. Mech. Anal. 187(1), (2008), 137–156.

[29] M. G. Crandall. Nonlinear Semigroups and Evolution Governed by Accretive Operators. In Proc. of Sympos. in Pure Mathematics, Part I, Vol. 45 ( F. Browder ed.). A.M.S., Providence 1986, pages 305–338.

[30] S. Dumont and N. Igbida. Back on a Dual Formulation for the Growing Sandpile Problem. Preprint.

[31] HI. Ekeland and R. Temam. Convex Analysis and Variational Problems. North-Holland, 1972.

[32] P. Fife. Some nonclassical trends in parabolic and parabolic-like evolutions. Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003.

[33] D. Kinderlehrer and G. Stampacchia. An Introduction to Variational Inequalities and their Applications, Academic Press, 1980.

[34] S. Kindermann, S. Osher and P. W. Jones. Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul., 4, (2005), 1091–1115.

[35] U. Mosco. Convergence of convex sets and solutions of variational inequalities. Advances. Math., 3 (1969), 510–585.

[36] L. Prigozhin. Variational models of sandpile growth. Europ. J. Appl. Math., 7 (1996), 225-236.

[37] L. Silvestre. Hölder estimates for solutions of integro differential equations like the fractional laplace. Indiana Univ. Math. J., 55(3), (2006), 1155–1174.

Fuensanta Andreu, Departament de Matemàtica Aplicada, Universitat de València Valencia, SPAIN.

E-mail address: fuensanta.andreu@uv.es

José M. Mazón and J. Julián Toledo-Melero Departament d’Anàlisi Matemàtica, Universitat de València Valencia, SPAIN.

E-mail address: mazon@uv.es, toledojj@uv.es

J. D. Rossi IMDEA Matematicas C-IX, Campus Cantoblanco UAM, Madrid, SPAIN. On leave from Dpto. de Matemáticas, FCEyN Universidad de Buenos Aires, 1428 Buenos Aires, ARGENTINA.

E-mail address: jrossi@dm.uba.ar