Elsevier

Nonlinear Analysis

Volume 212, November 2021, 112444
Nonlinear Analysis

Regularity of flat free boundaries for a p(x)-Laplacian problem with right hand side

https://doi.org/10.1016/j.na.2021.112444Get rights and content

Abstract

We consider viscosity solutions to a one-phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We apply the tools developed in De Silva (2011) to prove that flat free boundaries are C1,α. Moreover, we obtain some new results for the operator under consideration that are of independent interest.

Section snippets

Introduction and main results

In this paper we study a one-phase free boundary problem governed by the p(x)-Laplacian with non-zero right hand side. More precisely, we denote by Δp(x)udiv(|u|p(x)2u),where p is a function such that 1<p(x)<+. Then our problem is the following: Δp(x)u=f,in Ω+(u){xΩ:u(x)>0},|u|=g,on F(u)Ω+(u)Ω.Here ΩRn is a bounded domain, pC1(Ω), fC(Ω)L(Ω) and gC0,β(Ω), g0.

This problem comes out naturally from limits of a singular perturbation problem with forcing term as in [36], where the

Basic definitions, notation and preliminaries

In this section, we provide notation and basic definitions we will use throughout our work. We also present an auxiliary result on a Neumann problem that will be applied in the paper.

Notation

For any continuous function u:ΩRnR we denote Ω+(u){xΩ:u(x)>0},F(u)Ω+(u)Ω.We refer to the set F(u) as the free boundary of u, while Ω+(u) is its positive phase (or side).

Below we give the definition of viscosity solution to problem (1.1) and we deduce some consequences. In particular, we refer to the usual C

Different notions of solutions to p(x)-Laplacian

In this section we discuss the relationship between the different notions of solutions to Δp(x)u=f we are using, namely weak and viscosity solutions.

We start by observing that direct calculations show that, for C2 functions u such that u(x)0, Δp(x)u=div(|u|p(x)2u)=|u(x)|p(x)2Δu+(p(x)2)ΔNu+p(x),u(x)log|u(x)|,where ΔNuD2u(x)u(x)|u(x)|,u(x)|u(x)|denotes the normalized -Laplace operator.

First we need (see the Appendix for the definition of Sobolev spaces with variable

Auxiliary results

In this section we prove some results that will be of use in our main theorem. Namely, a Harnack inequality for an auxiliary problem of p(x)-Laplacian type and an existence result of barrier functions for the p(x)-Laplacian operator.

In the next result we assume for simplicity that fL(Ω)1, but a similar result holds for any fL(Ω). We have

Lemma 4.1

Assume that 1<pminp(x)pmax< with p(x) Lipschitz continuous in Ω and pLL, for some L>0. Let x0Ω and 0<R1 such that B4R(x0)¯Ω. Let vW1,p()(Ω)L

Geometric regularity results

In this section we prove a Harnack type inequality for a solution u to problem (1.1), following the approach in [15]. We will argue assuming that fL(Ω)ɛ2,g1L(Ω)ɛ2,pL(Ω)ɛ1+θ,pp0L(Ω)ɛ,holds, for 0<ɛ<1, for some constant 0<θ1.

The proof of Harnack inequality is based on the following lemma.

Lemma 5.1

Let u be a solution to (1.1)(5.1) in B1. There exists a universal constant ɛ̄ such that if 0<ɛɛ̄ and u satisfies q+(x)u(x)(q(x)+ɛ)+,xB1,q(x)=xn+σ,|σ|<120,and in x0=110en, u(x0)(q(x0)+ɛ2)+

Improvement of flatness

In this section we present the main improvement of flatness lemma. Theorem 1.1 will then be obtained by applying this lemma in an iterative way.

Lemma 6.1 Improvement of Flatness

Let u satisfy (1.1) in B1 and fL(B1)ɛ2,g1L(B1)ɛ2,pL(B1)ɛ1+θ,pp0L(B1)ɛ,for 0<ɛ<1, for some constant 0<θ1. Suppose that (xnɛ)+u(x)(xn+ɛ)+in B1,0F(u).If 0<rr0 for r0 universal, and 0<ɛɛ0 for some ɛ0 depending on r, then (xνrɛ/2)+u(x)(xν+rɛ/2)+in Br,with |ν|=1 and |νen|C̃ɛ for a universal constant C̃.

Proof

We divide the proof of

Regularity of the free boundary

In this section we finally prove our main result, namely, Theorem 1.1.

Proof of Theorem 1.1

Let u be a viscosity solution to (1.1) in B1 with 0F(u), g(0)=1 and p(0)=p0. Consider the sequence uk(x)=1ρku(ρkx),xB1,with ρk=r̄k, k=0,1,, for a fixed r̄ such that r̄β1/4,r̄r0,with r0 the universal constant in Lemma 6.1, taking θ=1 in (6.1).

Each uk is a solution to (1.1) with right hand side fk(x)=ρkf(ρkx), exponent pk(x)=p(ρkx), and free boundary condition gk(x)=g(ρkx). For the chosen r̄, by taking ɛ̄=ɛ0(r̄)2, the

Acknowledgment

The authors wish to thank Sandro Salsa for very interesting discussions about the subject of this paper.

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    1

    F. F. was partially supported by INDAM-GNAMPA, Italy 2019 project: Proprietà di regolarità delle soluzioni viscose con applicazioni a problemi di frontiera libera.

    2

    C. L. was partially supported by the project GHAIA Horizon 2020 MCSA RISE 2017 programme grant 777822 and by the grants CONICET, Argentina PIP 11220150100032CO 2016-2019, UBACYT, Argentina 20020150100154BA and ANPCyT, Argentina PICT 2016-1022. C. L. wishes to thank the Department of Mathematics of the University of Bologna, Italy, for the kind hospitality.

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