A Nonlocal Mean Curvature Flow

Consider a family { Γt}t≥0 of hypersurfaces embedded in \(\mathbb {R}^N\) parametrized by time t. Assume that each Γt = ∂Et, the boundary of a bounded open set Et in \(\mathbb {R}^N\).

Renormalized solutions for degenerate elliptic–parabolic problems with nonlinear dynamical boundary conditions and L1-data

Abstract We consider a degenerate elliptic–parabolic problem with nonlinear dynamical boundary conditions. Assuming L 1 -data, we prove existence and uniqueness in the framework of renormalized solutions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problems and in the weak formulation of the mathematical model of the so-called Hele–Shaw problem. Also, the problem with non-homogeneous Neumann boundary condition is included.

Stabilization of solutions of the filtration equation with absorption and non-linear flux

This paper is primarily concerned with the large time behaviour of solutions of the initial boundary value problem $$\begin{gathered} u_t = \Delta \phi (u) - \varphi (x,u)in\Omega \times (0,\infty ) \hfill \\ - \frac{{\partial \phi (u)}}{{\partial \eta }} \in \beta (u)on\partial \Omega \times (0,\infty ) \hfill \\ u(x,0) = u_0 (x)in\Omega . \hfill \\ \end{gathered} $$ Problems of this sort arise in a number of areas of science; for instance, in models for gas or fluid flows in porous media and for the spread of certain biological populations.

Optimal Mass Transport on Metric Graphs

We study an optimal mass transport problem between two equal masses on a metric graph where the cost is given by the distance in the graph. To solve this problem we find a Kantorovich potential as the limit of $p$-Laplacian--type problems in the graph where at the vertices we impose zero total flux boundary conditions. In addition, the approximation procedure allows us to find a transport density that encodes how much mass has to be transported through a given point in the graph, and also provides a simple formula of convex optimization for the total cost.

Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du)+γ(u) ∋ φ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x, Du)·η+β(u) ∋ ψ. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Δp(u) = div (|Du|p−2Du), with p > 1, γ and β are maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) and 0 ∈ β(0), and the data φ ∈ L1 (Ω) and ψ ∈ L1 (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with c…

Quasi-linear diffusion equations with gradient terms and L1 data

Abstract In this article we study the following quasi-linear parabolic problem: u t − Δ u+|u| β−2 u| ∇ u| q =|u| α−2 u| ∇ u| p in Ω×]0,T[, u(x,t)=0 on ∂Ω×]0,T[, u(x,0)=u 0 (x) in Ω, where Ω is a bounded open set of R N and T>0. We prove that if α,β>1, 0⩽p u 0 ∈L 1 (Ω) .

\( L^{1} \) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions

Abstract In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type − div a ( x , D u ) + γ ( u ) ∋ ϕ , posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a ( x , D u ) ⋅ η + β ( u ) ∋ ψ . The nonlinear elliptic operator div a ( x , D u ) is modeled on the p-Laplacian operator Δ p ( u ) = div ( | D u | p − 2 D u ) , with p > 1 , γ and β are maximal monotone graphs in R 2 such that 0 ∈ γ ( 0 ) and 0 ∈ β ( 0 ) , and the data ϕ ∈ L 1 ( Ω ) and ψ ∈ L 1 ( ∂ Ω ) .

Nonlocal Cheeger and Calibrable Sets

Given a non-null, measurable and bounded set \(\Omega \subset \mathbb {R}^N\), we define its J-Cheeger constant

A Monge-Kantorovich mass transport problem for a discrete distance

This paper is concerned with a Monge-Kantorovich mass transport problem in which in the transport cost we replace the Euclidean distance with a discrete distance. We fix the length of a step and the distance that measures the cost of the transport depends of the number of steps that is needed to transport the involved mass from its origin to its destination. For this problem we construct special Kantorovich potentials, and optimal transport plans via a nonlocal version of the PDE formulation given by Evans and Gangbo for the classical case with the Euclidean distance. We also study how these problems, when rescaling the step distance, approximate the classical problem. In particular we obta…

Fractional p-Laplacian evolution equations

Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for  x ∈ Ω ,  t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problem…

The Heat Content for Nonlocal Diffusion with Non-singular Kernels

Abstract We study the behavior of the heat content for a nonlocal evolution problem.We obtain an asymptotic expansion for the heat content of a set D, defined as ℍ D J ⁢ ( t ) := ∫ D u ⁢ ( x , t ) ⁢ 𝑑 x ${\mathbb{H}_{D}^{J}(t):=\int_{D}u(x,t)\,dx}$ , with u being the solution to u t = J ∗ u - u ${u_{t}=J\ast u-u}$ withinitial condition u 0 = χ D ${u_{0}=\chi_{D}}$ . This expansion is given in terms of geometric values of D. As a consequence, we obtain that ℍ D J ⁢ ( t ) = | D | - P J ⁢ ( D ) ⁢ t + o ⁢ ( t ) ${\mathbb{H}^{J}_{D}(t)=\lvert D\rvert-P_{J}(D)t+o(t)}$ as t ↓ 0 ${t\downarrow 0}$ .We also recover the usual heat content for the heat equation when we rescale the kernel J in an appro…

Porous medium equation with absorption and a nonlinear boundary condition

where is a bounded domain with smooth boundary, @=@ is the outer normal derivative, m ? 1; p; and q are positive parameters and u0 is in L∞( ). Problems of this form arise in mathematical models in a number of areas of science, for instance, in models for gas or :uid :ow in porous media [3] and for the spread of certain biological populations [13]. In the semilinear case (that is for m=1), there is an extensive literature about global existence and blow-up results for this type of problems, see among others, [5,9,16] and the literature therein. For the degenerate case (that is for m = 1), with a nonlinear boundary condition, local existence and uniqueness of weak solutions which are limit o…

On the best Lipschitz extension problem for a discrete distance and the discrete ∞-Laplacian

Abstract This paper concerns the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞-Laplacian problem, which arises as the dynamic programming formula for the value function of some e -tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p -Laplacian problem.

Optimal mass transportation for costs given by Finsler distances via p-Laplacian approximations

Abstract In this paper we approximate a Kantorovich potential and a transport density for the mass transport problem of two measures (with the transport cost given by a Finsler distance), by taking limits, as p goes to infinity, to a family of variational problems of p-Laplacian type. We characterize the Euler–Lagrange equation associated to the variational Kantorovich problem. We also obtain different characterizations of the Kantorovich potentials and a Benamou–Brenier formula for the transport problem.

Mass transport problems obtained as limits of p-Laplacian type problems with spatial dependence

Abstract. We consider the following problem: given a bounded convex domain Ω ⊂ ℝ N ${\Omega \subset \mathbb {R}^N}$ we consider the limit as p → ∞ of solutions to - div ( b p - p | D u | p - 2 D u ) = f + - f - ${- \operatorname{div} (b_{p}^{-p} |Du|^{p-2} Du)=f_+ - f_-}$ in Ω and b p - p | D u | p - 2 ∂ u ∂ η = 0 ${ b_{p}^{-p} |Du|^{p-2} \frac{\partial u}{\partial \eta }=0}$ on ∂ Ω ${\partial \Omega }$ . Under appropriate assumptions on the coefficients bp that in particular verify that lim p → ∞ b p = b ${ \lim _{p\rightarrow \infty } b_p = b }$ uniformly in Ω ¯ ${\overline{\Omega }}$ , we prove that there is a uniform limit of u p j ${u_{p_j}}$ (along a sequence p j → ∞ ${p_j \rightarrow…

Nonlocal Isoperimetric Inequality

For the nonlocal perimeter, there is also an isoperimetric inequality, and here the main hypothesis used on J is that it is radially nonincreasing.

Evolution problems of Leray-Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

Abstract In this paper we study evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in the framework of metric random walk spaces. This covers cases with the p -Laplacian operator in weighted discrete graphs and nonlocal operators with nonsingular kernel in R N .

Nonlocal Heat Content

The heat content of a Borel measurable set \(D \subset \mathbb {R}^N\) at time t is defined by M. van der Berg in [69] (see also [70]) as: $$\displaystyle \mathbb {H}_D(t) = \int _D T(t) {\chi }_D (x) dx, $$ with (T(t))t≥0 being the heat semigroup in \(L^2(\mathbb {R}^N)\). Therefore, the heat content represents the amount of heat in D at time t if in D the initial temperature is 1 and in \(\mathbb {R}^N \setminus D\) the initial temperature is 0.

Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source

Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of −g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra …

Existence and uniqueness for a degenerate parabolic equation with 𝐿¹-data

In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L 1 ( Ω ) L^{1}(\Omega ) , u t = d i v a ( x , D u ) in  ( 0 , ∞ ) × Ω , \begin{equation*}u_{t} = \mathrm {div} \mathbf {a} (x,Du) \quad \text {in } (0, \infty ) \times \Omega , \end{equation*} − ∂ u ∂ η a ∈ β ( u ) on  ( 0 , ∞ ) × ∂ Ω , \begin{equation*}-{\frac {{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text {on } (0, \infty ) \times \partial \Omega ,\end{equation*} u ( x , 0 ) = u 0 ( x ) in  Ω , \begin{equation*}u(x, 0) = u_{0}(x) \quad \text {in }\Omega ,\end{equation*} where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, ∂ / …

A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous t…

$(BV,L^p)$-decomposition, $p=1,2$, of Functions in Metric Random Walk Spaces

In this paper we study the $(BV,L^p)$-decomposition, $p=1,2$, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case $p=1$ we also study the associated geometric problem and the thresholding parameters.

OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

In this paper we study the questions of existence and uniqueness of solutions for equations of type - div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solut…

Nonlocal Minimal Surfaces and Nonlocal Curvature

Recall that if a set E has minimal local perimeter in a bounded set Ω, then it has zero mean curvature at each point of ∂E ∩ Ω (see [51]), and the equation that says that the curvature is equal to zero is the Euler–Lagrange equation associated to the minimization of the perimeter of a set.