Related papers: Almost Everywhere Regularity for the Free Boundary…

In this article we establish $C^{3,\alpha}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $\Omega \subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding…

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…

In this paper we study a one phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a C^1 surface in a neighborhood of every free boundary point. We also…

In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with…

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial…

We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric…

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result…

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right.…

This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{\lambda_1^s(\Omega)+\cdots+\lambda_m^s(\Omega) + \Lambda \mathcal{L}_n(\Omega)\colon \Omega\subset D \mbox{ s-quasi-open}\right\} $$ where $\Lambda>0,…

We establish a quasi-monotonicity formula {for an intrinsic frequency function related to solutions to} thin obstacle problems with zero obstacle driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$ bigger than the…

We prove $C^{2,\alpha}$ regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary…

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the…

In the optimal partial transport problem, one is asked to transport a fraction $0<m \leq \min\{||f||_{L^1}, ||g||_{L^1}\}$ of the mass of $f=f \chi_\Omega$ onto $g=g\chi_\Lambda$ while minimizing a transportation cost. If $f$ and $g$ are…

We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. In particular, viscosity solutions…

Let $\Omega\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary…

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_\Omega F(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap \Omega), \] where $F$ is a density function…

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class…

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit…

We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free…

This paper is devoted to a complete characterization of the free boundary of all solutions to the following spectral $k$-partition problem with measure and inclusion constraints: \[ \inf \left\{\sum_{i=1}^k \lambda_1(\omega_i)\; : \;…