arXiv subjects
Publications and source records attributed to Ovidiu Savin.
We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a noninfinitesimal notion of viscosity solutions.
For the Monge-Ampère equation with a right-hand side bounded away from 0 and infinity, we show that the solution, subject to the natural boundary condition arising in optimal transport, is in $W^{2,1+\varepsilon}$ up to the boundary.
We investigate global Hölder gradient estimates for solutions to the Monge-Ampère equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main situations when a) the domain $Ω$ is uniformly convex and b) $Ω$ is flat.
For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.
We study the equation $u_{11}u_{22} = 1$ in $\mathbb{R}^2$. Our results include an interior $C^2$ estimate, classical solvability of the Dirichlet problem, and the existence of non-quadratic entire solutions. We also construct global singular solutions to the analogous equation in higher dimensions. At the end we state some open questions.
In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale $r_0$ they satisfy a Weak Harnack type estimate. We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.
We consider Monge-Ampère equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $Ω$, which are of the type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;Ω,\quad\quad f\sim d^{-α}_{\partialΩ}\quad\mathrm{near}\;\partialΩ,$$ where $d_{\partialΩ}$ represents the distance to $\partial Ω$ and $-α$ is a negative power with $α\in(0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.
In this paper we extend the results of \cite{S3} to the borderline case $s = \frac 12$. We obtain the classification of global bounded solutions with asymptotically flat level sets for semilinear nonlocal equations of the type $$Δ^{\frac 12} u=W'(u) \quad \text{in} \quad R^n,$$ where $W$ is a double well potential.
We establish the optimal regularity for solutions to the multiple membranes problem, and perform a blow-up analysis at points on the free boundary with the highest multiplicity. This leads to a complete classification of blow-up profiles in the plane. The main technical tool is a Weiss-type monotonicity formula.
We consider the Bernoulli one-phase free boundary problem in a domain $Ω$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial Ω$. We achieve this by relating the behavior of $F$ near $\partial Ω$ to a Signorini-type obstacle problem.
We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations, as either two-plane solutions or solutions to a one-phase problem.
We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally $s$-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.
We give a definition of viscosity solution for the minimal surface system and prove a version of Allard regularity theorem in this setting.
We prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully nonlinear operators.
We obtain the classification of certain global bounded solutions for semilinear nonlocal equations of the type $$\triangle^s u=W'(u)$$ in $\mathbb{R}^n$,with $s \in (1/2 ,1),$ where $W$ is a double well potential.
We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the nonlinearity. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.
We simplify some technical steps from \cite{S1} in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type $$\triangle u=W'(u),$$ where $W$ is a double well potential.
We obtain $C^{2,β}$ estimates up to the boundary for solutions to degenerate Monge-Ampère equations of the type $$ \det D^2 u = f~~\text{in}~Ω, \quad \quad ~f\sim \text{dist}^α(\cdot, \partialΩ)~\text{near}~\partialΩ,~α>0. $$ As a consequence we obtain global $C^\infty$ estimates up to the boundary for the eigenfunctions of the Monge-Ampère operator $(\det D^2 u)^{1/n}$ on smooth, bounded, uniformly convex domains in $R^n$.