arXiv subjects

O. Savin

Publications and source records attributed to O. Savin.

4 recordsLinked to original sources

arXiv2016

The two membranes problem for different operators

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the H\"older continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain $C^{1,\gamma}$ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.

math.AP↗

arXiv2016

Obstacle type problems for minimal surfaces

We study certain obstacle type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.

math.AP↗

arXiv2009

Nonlocal minimal surfaces

The de Giorgi theory for minimal surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^σ$ norm, with $σ<1/2$, and try to understand what the minimisers look like. Parallel to the de Giorgi theory we prove that, if the boundary of a minimiser is sufficiently flat in the unit ball, then it is a smooth piece of hypersurface.

math.AP↗

arXiv2007

Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D

We prove that entire bounded monotone solutions to a certain class of fully nonlinear equations in 2D are one-dimensional. Our result also gives a new (non-variational) proof of the well known De Giorgi's conjecture.

math.AP↗