Related papers: Free boundary regularity in obstacle problems with…
This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $\Omega$; with $u$ and $v$ being both positive…
We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the…
In this paper, we prove Wiener's criterion for parabolic equations with singular and degenerate coefficients. To be precise, we study the problem of the regularity of boundary points for the Dirichlet problem for degenerate parabolic…
For the general obstacle problem, we prove by direct methods an epiperimetric inequality at regular and singular points, thus answering a question of Weiss (Invent. Math., 138 (1999), 23--50). In particular at singular points we introduce a…
We study the boundary behavior of solutions to parabolic double-phase equations through the celebrated Wiener's sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown…
We show the existence of a Lipschitz viscosity solution $u$ in $\Omega$ to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface $\partial \{ u> 0 \}\cap\Om$ and we show that the…
For the thin obstacle problem, we prove by a new direct method that in any dimension the Weiss' energies with frequency $\frac32$ and $2m$, for $m\in \mathbb N$, satisfy an epiperimetric inequality, in the latter case of logarithmic type.…
We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied…
In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns…
We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has…
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad…
We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional $I(u) = \int |\nabla u|^2 + V(u)$, where $V(u)$ is the characteristic function of the interval $(-1,1)$. This functional is a…
We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove the solutions are unique, and prove that the…
In this work, we investigate the continuity of the free boundary in a class of elliptic problems, with Neuman boundary condition. The main idea is a change of variable that allows us to reduce the problem to the one studied in [14].
We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity…
In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph-Legendre transformation we show that the regular part of the free boundary is real analytic. The first…
We consider a free boundary problem for the $p$-Laplace operator which is related to the so-called Bernoulli free boundary problem. In this formulation, the classical boundary gradient condition is replaced by a condition on the distance…
This paper investigates a class of $p$-obstacle problems with subcritical exponents having the form \begin{align} \mathrm{div}\left( a(x)|\nabla u|^{p-2}\nabla u\right) =m_1\chi_{\{u>0\}}-m_2u^{\lambda-1}\chi_{\{u>0\}} \ \text{in}\…
Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at…
We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and…