Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions
Raúl Ferreira Julio D. Rossi
Discrete & Continuous Dynamical Systems - A 2015, 35(4): 1469-1478 doi: 10.3934/dcds.2015.35.1469
In this paper we prove decay estimates for solutions to a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions, that is, $$u_t (x,t)=\int_{\Omega\cup\Omega_0} J(x-y) |u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\, dy$$ for $(x,t)\in \Omega\times \mathbb{R}^+$ and $u(x,t)=0$ in $ \Omega_0\times \mathbb{R}^+$. The proof of these estimates is based on bounds for the associated first eigenvalue.
keywords: eigenvalues. Nonlocal diffusion
Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1
Anna Mercaldo Julio D. Rossi Sergio Segura de León Cristina Trombetti
Communications on Pure & Applied Analysis 2013, 12(1): 253-267 doi: 10.3934/cpaa.2013.12.253
We consider the solution $u_p$ to the Neumann problem for the $p$--Laplacian equation with the normal component of the flux across the boundary given by $g\in L^\infty(\partial\Omega)$. We study the behaviour of $u_p$ as $p$ goes to $1$ showing that they converge to a measurable function $u$ and the gradients $|\nabla u_p|^{p-2}\nabla u_p$ converge to a vector field $z$. We prove that $z$ is bounded and that the properties of $u$ depend on the size of $g$ measured in a suitable norm: if $g$ is small enough, then $u$ is a function of bounded variation (it vanishes on the whole domain, when $g$ is very small) while if $g$ is large enough, then $u$ takes the value $\infty$ on a set of positive measure. We also prove that in the first case, $u$ is a solution to a limit problem that involves the $1-$Laplacian. Finally, explicit examples are shown.
keywords: $1$--Laplacian. Neumann problem $p$--Laplacian
Limits for Monge-Kantorovich mass transport problems
Jesus Garcia Azorero Juan J. Manfredi I. Peral Julio D. Rossi
Communications on Pure & Applied Analysis 2008, 7(4): 853-865 doi: 10.3934/cpaa.2008.7.853
In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, $\Omega$. Given two absolutely continuos measures (with respect to the surface measure) supported on the boundary $\partial \Omega$, by performing a suitable extension of the measures to a strip of width $\varepsilon$ near the boundary of the domain $\Omega$ we consider the mass transfer problem for the extensions. Then we study the limit as $\varepsilon$ goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the $p-$Laplacian for large values of $p$.
keywords: quasilinear elliptic equations Mass transport Neumann boundary conditions.
Stability of the blow-up time and the blow-up set under perturbations
José M. Arrieta Raúl Ferreira Arturo de Pablo Julio D. Rossi
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 43-61 doi: 10.3934/dcds.2010.26.43
In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems.
   We also present several examples. Among them we consider changing the spacial domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow-up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set.
keywords: blow-up Stability perturbations.
Tug-of-war games and the infinity Laplacian with spatial dependence
Ivana Gómez Julio D. Rossi
Communications on Pure & Applied Analysis 2013, 12(5): 1959-1983 doi: 10.3934/cpaa.2013.12.1959
In this paper we look for PDEs that arise as limits of values of tug-of-war games when the possible movements of the game are taken in a family of sets that are not necessarily Euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form $- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$, that is, an infinity Laplacian with spatial dependence. Here $J_x (Dv(x))$ is a vector that depends on the spatial location and the gradient of the solution.
keywords: viscosity solutions Tug-of-war games infinity Laplacian.
Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains
Julián Fernández Bonder Julio D. Rossi
Communications on Pure & Applied Analysis 2002, 1(3): 359-378 doi: 10.3934/cpaa.2002.1.359
We study the asymptotic behavior for the best constant and extremals of the Sobolev trace embedding $W^{1,p} (\Omega) \rightarrow L^q (\partial \Omega)$ on expanding and contracting domains. We find that the behavior strongly depends on $p$ and $q$. For contracting domains we prove that the behavior of the best Sobolev trace constant depends on the sign of $qN-pN+p$ while for expanding domains it depends on the sign of $q-p$. We also give some results regarding the behavior of the extremals, for contracting domains we prove that they converge to a constant when rescaled in a suitable way and for expanding domains we observe when a concentration phenomena takes place.
keywords: Sobolev trace constants p-Laplacian eigenvalue problems. nonlinear boundary conditions
A logistic equation with refuge and nonlocal diffusion
J. García-Melián Julio D. Rossi
Communications on Pure & Applied Analysis 2009, 8(6): 2037-2053 doi: 10.3934/cpaa.2009.8.2037
In this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\lambda$.
keywords: Nonlocal diffusion logistic problems.
Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary
C. Brändle F. Quirós Julio D. Rossi
Communications on Pure & Applied Analysis 2005, 4(3): 523-536 doi: 10.3934/cpaa.2005.4.523
We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as non-simultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.
keywords: nonlinear diffusion parabolic systems nonlinear boundary conditions. Blow-up
A convex-concave elliptic problem with a parameter on the boundary condition
J. García-Melián Julio D. Rossi José Sabina de Lis
Discrete & Continuous Dynamical Systems - A 2012, 32(4): 1095-1124 doi: 10.3934/dcds.2012.32.1095
In this paper we study existence and multiplicity of nonnegative solutions to $$ \begin{equation} \left\{\begin{array}{ll} \Delta u = u^p + u^q \qquad & \mbox{in }\Omega, \\ \frac{\partial u }{\partial \nu} =\lambda u \qquad & \mbox{on }\partial \Omega. \end{array}\right. \end{equation} $$ Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\nu$ stands for the outward unit normal and $p$, $q$ are in the convex-concave case, that is $0 < q < 1 < p$. We prove that there exists $\Lambda^* >0$ such that there are no nonnegative solutions for $\lambda < \Lambda^*$, and there is a maximal nonnegative solution for $\lambda \ge \Lambda^{*}$. If $\lambda$ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when $\lambda\to \infty$ and the occurrence of dead cores. In the particular case where $\Omega$ is the unit ball of $\mathbb{R}^N$ we show exact multiplicity of radial nonnegative solutions when $\lambda$ is large enough, and also the existence of nonradial nonnegative solutions.
keywords: positive solutions Convex-concave multiplicity.
An obstacle problem for Tug-of-War games
Juan J. Manfredi Julio D. Rossi Stephanie J. Somersille
Communications on Pure & Applied Analysis 2015, 14(1): 217-228 doi: 10.3934/cpaa.2015.14.217
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
keywords: Obstacle Problem infinity laplacian. Tug-of-War games

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