arXiv subjects
Publications and source records attributed to Sandra Martinez.
In this paper we construct two algorithms to approximate the minimizer of a discrete functional which comes from using a discontinuous Galerkin method for a variational problem related to the $p(x)$-Laplacian. We also make some numerical experiments in dimension two.
In this paper we study the $H^2$ global regularity for solutions of the $p(x)-$Laplacian in two dimensional convex domains with Dirichlet boundary conditions. Here $p:Ω\to [p_1,\infty)$ with $p\in Lip(\barΩ)$ and $p_1>1$.
In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$ a solution of, $$ Łu^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla \uep\Big)=β_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$ A solution to $(P_{\ep})$ is a function $u^{\ep}\in W^{1,G}(Ω)\cap L^{\infty}(Ω)$ such that $$ \int_Ω g(|\nabla u^{\ep}|) \frac{\nabla u^{\ep}}{|\nabla u^{\ep}|} \nabla ϕdx =-\int_Ω ϕβ_{\ep}(u^{\ep}) dx $$ for every $ϕ\in C_0^{\infty}(Ω)$. Here $β_{\ep}(s)= \frac{1}{\ep} β(\frac{s}{\ep}), $ with $β\in {Lip}(\R)$, $β>0$ in $(0,1)$ and $β=0$ otherwise. We are interested in the limiting problem, when $\ep\to 0$. As in previous work with $Ł=Δ$ or $Ł=Δ_p$ we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a $C^{1,α}$ surface. This result is new even for $Δ_p$. Throughout the paper we assume that $g$ satisfies the conditions introduced by G. Lieberman in \cite{Li1}
We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|)+λχ_{\{u>0\}} dx$ in the class of functions $W^{1,G}(Ω)$ with $u-ϕ_0\in W_0^{1,G}(Ω)$, for a given $ϕ_0\geq 0$ and bounded. $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|) dx<\infty$. The conditions on the function G allow for a different behavior at 0 and at $\infty$. We prove that every solution u is locally Lipschitz continuous, that they are solution to a free boundary problem and that the free boundary, $\partial\{u>0\}\cap Ω$, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the $C^{1,α}$ regularity of their free boundaries near ``flat'' free boundary points.
We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|) dx$ in the class of functions $W^{1,G}(Ω)$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\infty$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap Ω$, is smooth.
We consider the optimization problem of minimizing $\int_Ω|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap Ω$, is smooth.