Related papers: Interface regularity for semilinear one-phase prob…
In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the…
In this paper, we consider a free boundary problem of a semilinear nonhomogeneous elliptic equation with Bernoulli's type free boundary. The existence and regularity of the solution to the free boundary problem are established by use of the…
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 consider a one-phase Bernoulli free boundary problem in a container $D$ - a smooth open subset of $\mathbb{R}^d$ - under the condition that on the fixed boundary $\partial D$ the normal derivative of the solutions is prescribed. We study…
We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply-connected. We show that if two components of the free boundary are…
We consider a one-phase free boundary problem with variable coefficients and non-zero right hand side. We prove that flat free boundaries are $C^{1,\alpha}$ using a different approach than the classical supconvolution method of Caffarelli.…
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. This completes the study of…
We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an existence and a…
In this paper, we investigate Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces. We prove the existence of minimizers and prove the local Lipschitz continuity of minimizers provided that the negative part is locally…
We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function strictly increasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. Our main result…
In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are $C^{1,\alpha}$. The method consists of employing a hodograph transform and deriving the regularity via a linearization technique,…
In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase…
We show that any minimizer of the well-known ACF functional (for the $p$-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down…
We study global solutions to the classical one-phase free boundary problem that have finite Morse index relative to the Alt-Caffarelli functional. We show that such solutions are stable outside a compact set and characterize the index as…
In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove…
Using a direct approach, we prove a $2$-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension $2$, the $C^{1,\alpha}$ regularity…
We study the structure of solutions of the interior Bernoulli free boundary problem for $(-\Delta)^{\alpha/2}$ on an interval $D$ with parameter $\lambda > 0$. In particular, we show that there exists a constant $\lambda_{\alpha,D} > 0$…
We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein,…
In this article we study for the first time the regularity of the free boundary in the one-phase free boundary problem driven by a general nonlocal operator. Our main results establish that the free boundary is $C^{1,\alpha}$ near regular…
We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal…