Regularity of flat free boundaries for a -Laplacian problem with right hand side
Section snippets
Introduction and main results
In this paper we study a one-phase free boundary problem governed by the -Laplacian with non-zero right hand side. More precisely, we denote by where is a function such that . Then our problem is the following: Here is a bounded domain, , and , .
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 we denote We refer to the set as the free boundary of , while 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
Different notions of solutions to -Laplacian
In this section we discuss the relationship between the different notions of solutions to we are using, namely weak and viscosity solutions.
We start by observing that direct calculations show that, for functions such that , where 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 -Laplacian type and an existence result of barrier functions for the -Laplacian operator.
In the next result we assume for simplicity that , but a similar result holds for any . We have
Lemma 4.1 Assume that with Lipschitz continuous in and , for some . Let and such that . Let
Geometric regularity results
In this section we prove a Harnack type inequality for a solution to problem (1.1), following the approach in [15]. We will argue assuming that holds, for , for some constant 1.
The proof of Harnack inequality is based on the following lemma.
Lemma 5.1 Let be a solution to (1.1)–(5.1) in . There exists a universal constant such that if and satisfies and in ,
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 satisfy (1.1) in and for , for some constant . Suppose that If for universal, and for some depending on , then with and for a universal constant .
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 be a viscosity solution to (1.1) in with , and . Consider the sequence with , , for a fixed such that with the universal constant in Lemma 6.1, taking in (6.1). Each is a solution to (1.1) with right hand side , exponent , and free boundary condition . For the chosen , by taking , the
Acknowledgment
The authors wish to thank Sandro Salsa for very interesting discussions about the subject of this paper.
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F. F. was partially supported by INDAM-GNAMPA, Italy 2019 project: Proprietà di regolarità delle soluzioni viscose con applicazioni a problemi di frontiera libera.
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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.