Related papers: Free boundary regularity for semilinear variationa…
In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$,…
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump L\'evy processes. In this paper we study the regularity of the free boundary. Our main…
We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers,…
Given~$s,\sigma\in(0,1)$ and a bounded domain~$\Omega\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $\sigma$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(\Omega^c)^2) } +…
In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz and Shahgholian investigated the regularity of the solution to the obstacle problem with singular logarithmic forcing term \begin{equation*} -\Delta u = \log u \,…
The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions.
It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the…
While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…
In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim \chi_{\{u > 0 \}}$. The…
In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an…
We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that…
A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form $$\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}|\nabla…
In this paper we consider a weakly coupled $p$-Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free…
The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular…
We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the…
In this paper we are concerned with singular points of solutions to the {\it unstable} free boundary problem $$ \Delta u = - \chi_{\{u>0\}} \qquad \hbox{in} B_1. $$ The problem arises in applications such as solid combustion, composite…
Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the…
We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting,…
We study free boundary problems for incompressible inhomogeneous flows governed by the Navier--Stokes equations, focusing on the regularity and global-in-time well-posedness of solutions in critical functional frameworks for small initial…