Related papers: Some remarks on stability of cones for the one-pha…
The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions $n\geq3$ is completely open. In…
In this paper we prove a classification result for axially symmetric one phase minimizers of the Alt-Phillips free boundary problem in dimensions 3, 4, and 5. To accomplish this, we establish a stability inequality that extends the one for…
We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension $7$ and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence,…
Given an axially-symmetric, $(n+1)$-dimensional convex cone $\Omega\subset \mathbb{R}^{n+1}$, we study the stability of the free-boundary minimal surface $\Sigma$ obtained by intersecting $\Omega$ with a $n$-plane that contains the axis of…
We study minimizing singular cones with free boundary associated with the capillarity problem. Precisely, we provide a stability criterion $\`a$ la Jerison-Savin for capillary hypersurfaces and show that, in dimensions up to $4$, minimizing…
We prove that a stable $C^{1,1}$-to-edge properly embedded free boundary minimal hypersurface $\Sigma^3$ of a $4$-dimensional wedge domain $\Omega^4_{\theta}$ with angle $\theta\in (0,\pi]$ is flat.
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the…
For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the…
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.
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,…
For each sufficiently large integer $k$, we construct a domain in the round $2$-sphere with $k$ boundary components which is the link of a cone in $\mathbb{R}^3$ admitting a homogeneous solution to the one-phase free boundary problem. This…
We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and…
We study solutions to a one-phase singular perturbation problem that arises in combustion theory and that formally approximates the classical one-phase free boundary problem. We introduce a natural density condition on the transition layers…
We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such…
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…
Roughly speaking, a conic bundle is a surface, fibered over a curve, such that the fibers are conics (not necessarily smooth). We define stability for conic bundles and construct a moduli space. We prove that (after fixing some invariants)…
In this paper, we are interested in bounded, entire, solutions of the Allen-Cahn equation which are defined in Euclidean space and whose zero set is asymptotic to a given minimal cone. In particular, in dimension larger than or equal to 8,…
We study the free boundary of the porous medium equation with nonlocal drifts in dimension one. Under the assumption that the initial data has super-quadratic growth at the free boundary, we show that the solution is smooth in space and…
We study the regularity of the free boundary in one-phase Stefan problem with nonlinear operator. Using the Hodograph transform and a linearization technique, we prove that flat free boundaries are $C^{1, \alpha}$ in space and time. When…
Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a…