No abstract

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζ ε (u) with ζ ε → δ 0 · ζ . We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F (D 2 u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.

2016-11-03T14:11:40

No abstract

We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.Keywords: Parabolic fully nonlinear equations, singularly perturbed problems, Lipschitz regularity, porosity of the free boundary.

We establish sharp geometric C 1+α regularity estimates for bounded weak solutions of evolution equations of p-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in [DeS] to prove that flat free boundaries and Lipschitz free boundaries are C 1,γ . keywords: free boundary problems, degenerate fully non-linear elliptic operators, regularity theory.

In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when “the gradient is small”) under strong absorption conditions of the general form: where the mapping fails to decrease fast enough at the origin, so allowing that nonnegative solutions may create plateau regions, that is, a priori unknown subsets where a given solution vanishes identically. We establish improved geometric regularity along the set (the free boundary of the model), for a sharp value of (obtained explicitly) depending only on structural parameters. Non‐degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.