arXiv subjects
Publications and source records attributed to Ovidiu Savin.
We study minimizing cones in the Alt-Phillips problem when the exponent γ is close to 1. When γ converges to 1, we show that the cones concentrate around symmetric solutions to the classical obstacle problem. To be precise, the limiting profiles are radial in a subspace and invariant in directions perpendicular to that subspace.
We establish global $C^{1,β}$ and $W^{2, p}$ regularity for singular Monge-Ampère equations of the form \[\det D^2 u \sim \text{dist}^{-α}(\cdot,\partialΩ),\quad α\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation \[\det D^2 u=|u|^{-α}\quad \text{in}\quadΩ,\quad u=0\quad \text{in}\quad \partialΩ, \quad α\in (0, 1),\] where $Ω$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,β}$ and belongs to $W^{2, p}$ for all $p<1/α$.
We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,γ}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.
We provide the classical Boundary Harnack principle in Lipschitz domains for solutions to two different linear uniformly elliptic equations with the same principal part.
We consider a class of Allen-Cahn equations associated with Ginzburg-Landau energies involving degenerate double-well potentials that vanish of order $m$ at the minima \begin{equation} J(v,\Omega)=\int_{\Omega}\Big\{|\nabla v|^{p}+(1-v^{2})^{m}\Big\}dx,\quad 1<p<m, \end{equation} and establish density estimates for the level sets of nontrivial minimizers $|v| \leq 1$. This extends a result of Dipierro-Farina-Valdinoci where the density estimates for such degenerate potentials were obtained for a bounded range of $m$'s. The original estimates for the classical case $p=m=2$ were established by Caffarelli-C\'ordoba.
For all odd positive integers $m$, we construct $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^3,$ with $μ\in(m,m+1)$. For $m$ large, $μ-m$ converges to $1$, so $μ\neq m+\tfrac 1 2$. The restriction to odd values of $m$ is necessary: we show that, for all $n\ge 2$, there are no $μ$-homogeneous solutions to the thin obstacle problem in $\mathbb{R}^n$ with $μ\in \bigcup_{k\ge 0}(2k,2k+1)$. These examples also apply to $2$-valued $C^{1,1/2}$ stationary harmonic functions or $\mathbb{Z}/2\mathbb{Z}$-eigenfunctions of the laplacian on the sphere.
We establish Liouville theorems for global minimizers $u$ of the Allen-Cahn energy $$\int |\nabla u|^2 + W(u) \, dx,$$ which have subquadratic growth at infinity. In particular we extend the results of \cite{S1,S3} concerning the De Giorgi's conjecture to the setting of unbounded solutions. Part of the analysis relies on the regularity of minimizers for a Dirichlet/perimeter functional which was studied by Athanasopoulous-Caffarelli-Kenig-Salsa in \cite{ACKS}.
We study homogeneous solutions to the Alt-Phillips problem when the exponent $γ$ is close to 1. In dimension $d\ge3$, we show that the radial cone is minimizing when $γ$ is close to 1. In dimension $d \ge 4$, we construct an axially symmetric cone whose contact set has with positive density. We show that it is a global minimizer. It is analogous to the De Silva-Jerison \cite{DJ} cone for the Alt-Caffarelli functional which corresponds to exponent $γ=0$. The cone we construct bifurcates from another minimizing cone whose contact set has zero density, obtained as the trivial extension of the radial solution. This second cone is analogous to a quadratic polynomial solution in the classical obstacle problem which corresponds to exponent $γ=1$. In particular our results show that, when $γ<1$ is sufficiently close to 1, there are axis symmetric cones that exhibit the properties of both end point cases $γ=0$ and $γ=1$.
We give a simple proof of the fact that - in all dimensions - there are no homogeneous solutions to the thin obstacle problem with frequency $λ$ belonging to intervals of the form $(2k,2k+1)$, $k \in \mathbb{N}$. In particular, there are no frequencies in the interval $(2,3)$.
In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious, since the surfaces may touch at an irregular point, therefore a suitable blow-up analysis must be combined with a bespoke regularity theory to obtain this result. For the classical case, an analogous result was proved by Leon Simon. Our proof also relies on a Harnack Inequality for nonlocal minimal surfaces that has been recently introduced by Xavier Cabré and Matteo Cozzi and which can be seen as a fractional counterpart of a classical result by Enrico Bombieri and Enrico Giusti. In our setting, an additional difficulty comes from the analysis of the corresponding nonlocal integral equation on a hypersurface, which presents a remainder whose sign and fine properties need to be carefully addressed.
Lawson and Osserman proved that the Dirichlet problem for the minimal surface system is not always solvable in the class of Lipschitz maps. However, it is known that minimizing sequences (for area) of Lipschitz graphs converge to objects called Cartesian currents. Essentially nothing is known about these limits. We show that such limits can have surprisingly large interior vertical and non-minimal portions. This demonstrates a striking discrepancy between the parametric and non-parametric area minimization problems in higher codimension. Moreover, our construction has the smallest possible dimension ($n = 3$) and codimension $(m = 2)$.
We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set $K \subset \mathbb{R}^n$ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \cite{C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain $K$. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest.
We construct viscosity solutions to the special Lagrangian equation that are Lipschitz but not $C^1$.
Differently from their classical counterpart, nonlocal minimal surfaces are known to present boundary discontinuities, by sticking at the boundary of smooth domains. It has been observed numerically by J. P. Borthagaray, W. Li, and R. H. Nochetto ``that stickiness is larger near the concave portions of the boundary than near the convex ones, and that it is absent in the corners of the square'', leading to the conjecture ``that there is a relation between the amount of stickiness on $\partialΩ$ and the nonlocal mean curvature of $\partialΩ$''. In this paper, we give a positive answer to this conjecture, by showing that the nonlocal minimal surfaces are continuous at convex corners of the domain boundary and discontinuous at concave corners. More generally, we show that boundary continuity for nonlocal minimal surfaces holds true at all points in which the domain is not better than $C^{1,s}$, with the singularity pointing outward, while, as pointed out by a concrete example, discontinuities may occur at all point in which the domain possesses an interior touching set of class $C^{1,α}$ with $α>s$.
We consider an energy model for harmonic graphs with junctions and study the regularity properties of minimizers and their free boundaries.
We investigate the rigidity of global minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2),$$ when the exponent $γ$ is close to the extremes of the admissible values. In particular we show that global minimizers in $\mathbb{R}^n$ are one-dimensional if $γ$ is close to 2 and $n \le 7$, or if $γ$ is close to $0$ and $n \le 4$.
We study the most common image and informal description of the optimal transport problem for quadratic cost, also known as the second boundary value problem for the Monge--Ampère equation -- What is the most efficient way to fill a hole with a given pile of sand? -- by proving regularity results for optimal transports between degenerate densities. In particular, our work contains an analysis of the setting in which holes and sandpiles are represented by absolutely continuous measures concentrated on bounded convex domains whose densities behave like nonnegative powers of the distance functions to the boundaries of these domains.
We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$$ considered by Athanasopoulous, Caffarelli, Kenig, and Salsa.