Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the $p(x)$-Laplacian with non-zero right hand side. We prove that flat free boundaries are $C^{1,\gamma}$. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the $p(x)$-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when $p(x)\equiv p$, i.e., for the $p$-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability

A singular perturbation problem for the p(x)-Laplacian

We present results for the following singular perturbation problem: ∆p(x)uε := div(|∇uε(x)| p(x)−2∇uε) = βε(uε) + f ε, uε ≥ 0 (Pε(f ε)) in Ω ⊂ RN , where ε > 0, βε(s) = 1 εβ( s ε ), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and β(s) ds = M. The functions uε and f ε are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ε → 0) and we show that limit functions are weak solutions to a free boundary problem.Fil: Lederman, Claudia Beatriz. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Wolanski, Noemi Irene. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Lipschitz continuity of minimizers in a problem with nonstandard growth

In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional J(v) = ∫ Ω - F(x; v; ∇v) + (x)νfv>0) dx, under nonstandard growth conditions of the energy function F(x; s; η) and 0 < λmin ≤ λ (x) ≤ λmax < 1. This is the optimal regularity for the problem. Our results generalize the ones we obtained in the case of the inhomogeneous p(x)-Laplacian in our previous work. Nonnegative local minimizers u satisfy in their positivity set a general nonlinear degenerate/singular equation divA(x; u; ∇u) = B(x; u; ru) of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.Fil: Lederman, Claudia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Recent Results on Nonlinear Elliptic Free Boundary Problems

In this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.Fil: Ferrari, Fausto. Universidad de Bologna; ItaliaFil: Lederman, Claudia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Salsa, Sandro. Politecnico di Milano; Itali

Uniqueness in a two phase free boundary problem

We investigate a two-phase free-boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u(x, t) defined in a domain D ⊂ RN × (0, T) which satisfies the equation ∆u + Σ ai uxi − ut = 0 whenever u(x, t) = 0, i.e., in the subdomains D+ = {(x, t) ∈ D : u(x, t) > 0} and D− = {(x, t) ∈ D : u(x, t) 0 is a fixed constant, and the gradients are spatial sidederivatives in the usual two-phase sense. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of D. The problem admits classical solutions only for good data and for small times. To overcome this problem several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. Continuing the work done for the one-phase problem we investigate conditions under which the three concepts agree and produce a unique solution for the two-phase problem.Fil: Lederman, Claudia Beatriz. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Vazquez, Juan Luis. Universidad Autonoma de Madrid; EspañaFil: Wolanski, Noemi Irene. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Uniqueness of solution to a free boundary problem from combustion

We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function u(x, t) ≥ 0, defined in a domain D ⊂ RN × (0, T) and such that ∆u +Xai uxi − ut = 0 in D∩{u > 0}. We also assume that the interior boundary of the positivity set, D ∩ ∂{u > 0}, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: u = 0, −∂u/∂ν = C. Here ν denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of D. This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.Fil: Lederman, Claudia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Vázquez, Juan Luis. Universidad Autónoma de Madrid; EspañaFil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin