Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources

Sandro Salsa,Gianmaria Verzini,Francesco Tulone

doi:10.3934/mine.2018.1.147

Sandro Salsa, Gianmaria Verzini + Show 1 more

Open Access

https://doi.org/10.3934/mine.2018.1.147

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Abstract

In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Caffarelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side.

Highlights

In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Caffarelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side
In the last years the regularity theory for two phase problems governed by uniformly elliptic equations with distributed sources has reached a considerable level of completeness extending the results in the seminal papers [2, 4] and in [17, 18] to the inhomogeneous case, through a different approach first introduced in [7]
Existence of a continuous viscosity solution through a Perron method has been established for linear operators in divergence form in [3] and in [9], and for a class of concave operators in [19]

Summary

Introduction

In the last years the regularity theory for two phase problems governed by uniformly elliptic equations with distributed sources has reached a considerable level of completeness (see for instance the survey paper [10]) extending the results in the seminal papers [2, 4] (for the Laplace operator) and in [17, 18] (for concave fully non linear operators) to the inhomogeneous case, through a different approach first introduced in [7]. In particular the papers [15] and [8] provides optimal Lipschitz regularity for viscosity solutions and their free boundary for a large class of fully nonlinear equations. We introduce our class of free boundary problems and their weak (or viscosity) solutions. Ν = ν(x) denotes the unit normal to the free boundary F = F (u) at the point x, pointing toward Ω+, while the function G(β, x, ν) is Lipschitz continuous, strictly increasing in β, and inf G(0, x, ν) > 0. The main aim of this paper is to adapt Perron’s method in order to prove the existence of a weak (viscosity) solution of the above f.b.p., with assigned Dirichlet boundary conditions For any u continuous in Ω we say that a point x0 ∈ F (u) is regular from the right C and so on will be termed “universal” if they only depend on λ, Λ, n, Ω, fi ∞ and g

Asymptotic developments

The function u is Lipschitz continuous

The function u is a supersolution

The function u is a subsolution

Properties of the free boundary

Some explicit barrier functions