Related papers: A capillarity one-phase Bernoulli free boundary pr…
Consider the two-phase free boundary problem subject to surface tension and gravitational forces for a class of non-Newtonian fluids with stress tensors $T_i$ of the form $T_i=-\pi I+\mu_i(|D(v)|^2)D(v)$ for $i=1,2$, respectively, and where…
In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in $\mathbb{R}^2$. We establish a prior gradient estimate for smooth solutions to this…
In this paper, we analyze the singular shape of the free boundary at degenerate points in a three dimensional axisymmetric compressible gravity flow. For all possible degenerate points on the free surface, we prove that the only nontrivial…
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points…
We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…
We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain $\Omega^{\varepsilon}(t).$ Under suitable conditions on the given data, the one-valued local classical solvability of the problem for…
We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are H\"older continuous and the free boundary has…
We study pure $D$ dimensional Einstein gravity in spacetimes with a generic null boundary. We focus on the symplectic form of the solution phase space which comprises a $2D$ dimensional boundary part and a $2(D(D-3)/2+1)$ dimensional bulk…
We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately…
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 consider the problem of optimal partition of a domain with respect to the sum of the principal eigenvalues and we prove for the first time regularity results for the free interface up to fixed boundary. All our results are quantitative…
In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{cases} F(D^2u)-\partial_tu=u^a\chi_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on…
This paper is concerned with the long time dynamics of the free boundary of a Darcy fluid in three space dimensions, also known as the one-phase Muskat problem. The dynamics of the free boundary is governed by a nonlocal fully nonlinear…
This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume…
We consider the scaling limit of a generic ferromagnetic system with a continuous phase transition, on the half plane with boundary conditions leading to the equilibrium of two different phases below criticality. We use general properties…
We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of the solutions and some geometric…
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 the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our…
In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition…
In this paper we consider a free boundary problem in the 3-dimensional Lorentz-Minkowski space $\l^3$ which deals spacelike surfaces whose mean curvature is a linear function of the time coordinate and the boundary moves in a given support…