Free boundary regularity for a degenerate problem with right hand side

Leitão, Raimundo; Ricarte, Gleydson C.

doi:10.4171/ifb/413

Cited by 11 publications

(5 citation statements)

References 4 publications

(4 reference statements)

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“…We are interested in the regularity of the free boundary for viscosity solutions of (18), following the strategy introduced in [22]. The same technique was applied to the p-Laplace operator (p(x) ≡ p in ( 18)), with p ≥ 2, in [56].…”

Section: Nonlinear Operators With Non-standard Growthmentioning

confidence: 99%

See 1 more Smart Citation

Recent Results On Nonlinear Elliptic Free Boundary Problems

Ferrari¹,

Lederman²,

Salsa³

2022

Preprint

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Section: Nonlinear Operators With Non-standard Growthmentioning

confidence: 99%

“…Another essential tool leading to the proof of Theorem 17 is given by the following Harnack type inequality, obtained in [38], whose p−Laplace version (i.e., for p(x) ≡ p constant) and with p ≥ 2, is contained in [56].…”

Section: Nonlinear Operators With Non-standard Growthmentioning

confidence: 99%

Recent Results On Nonlinear Elliptic Free Boundary Problems

Ferrari¹,

Lederman²,

Salsa³

2022

Preprint

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show abstract

“…Thus, even at the boundary, a viscosity interpretation seems to be the most convenient one in order to manage both existence and uniqueness questions. More precisely, throughout the paper we interpret solutions to (P ) λ to mean viscosity solutions defined in the next definition, introduced by De Silva [28, Definitions 2.2 and 2.3], and has been adopted in several subsequent works (see for instance [29,45,46]). If u, v : Ω → R are two functions and x ∈ Ω, by u ≺ x v we mean that u(x) = v(x) and u(y) ≤ v(y) in a neighborhood of x.…”

Section: Notion Of Solutionmentioning

confidence: 99%

Bernoulli Free Boundary Problem for the Infinity Laplacian

Crasta¹,

Fragalà²

2020

SIAM J. Math. Anal.

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“…In [Caf87,Caf89,Caf88] Caffarelli developed a viscosity approach to the free boundary problem for p = 2 in the two-phase setting (see also [DS11] and the references therein for related free boundary regularity properties). For one-phase problems the viscosity approach for operators governed by the p-Laplace or p(x)−Laplace operators has been developed in [LR18,FL21] and in [FL22] as well, where Lipschitz regularity of viscosity solutions of inhomogeneous one-phase problems, governed by the p(x)−Laplace operator and some regularity properties of their free boundaries were established.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz regularity of almost minimizers in one-phase problems driven by the $p$-Laplace operator

Dipierro¹,

Ferrari²,

Nicolò³

et al. 2022

Preprint

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We prove that, given p > max 2n n+2 , 1 , the nonnegative almost minimizers of the nonlinear free boundary functional J p (u, Ω) := ˆΩ |∇u(x)| p + χ {u>0} (x) dx are Lipschitz continuous.The setting of almost minimizers is also classical in the calculus of variations (dating back, at least up to a certain extent, to a famous sentence in Leibniz's Specimen Geometriae Luciferae, probably written in the mid-1690s, "pro minimis adhiberi possunt quasi minima", that is "the almost minimizers can be exploited in place of minimizers").More specifically, the mathematical setting that we consider here goes as follows. Let Ω ⊂ R n be a given domain and p > max 2n n+2 , 1 . We consider the energy functionalThe condition that u is nonnegative corresponds, in the framework of free boundary problems, to considering "one-phase" solutions (solutions which may change sign being related to "twophase" problems).The precise notion of almost minimizers that we use in this paper is the following one: Definition 1.1. Let κ 0 and β > 0. We say that u ∈ W 1,p (Ω) is an almost minimizer for J p in Ω, with constant κ and exponent β, if u 0 a.e. in Ω andfor every ball B ̺ (x) such that B ̺ (x) ⊂ Ω and for every v ∈ W 1,p (B ̺ (x)) such that v = u on ∂B ̺ (x) in the sense of the trace.In some sense, Definition 1.1 is one of the possible modern formalizations of Leibniz's initial intuition reported at the beginning of this paper: namely, almost minimizers are natural objects to look at, for instance, to deal with minimizers of "perturbed" functionals. As a concrete example, if we considerfor a function Φ : R → [0, 1] with Φ = 0 in (−∞, 0], we readily see that J p (u, B r (x)) J p (u, B r (x)) andAccordingly, a minimizer for the "complicated" functional J p turns out to be an almost minimizer for the "simpler" functional J p .As usual, the constants depending only on n and p are called universal. If u is an almost minimizer, the structural constants may depend on κ and β as well.

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Free boundary regularity for a degenerate problem with right hand side

Cited by 11 publications

References 4 publications

Recent Results On Nonlinear Elliptic Free Boundary Problems

Recent Results On Nonlinear Elliptic Free Boundary Problems

Bernoulli Free Boundary Problem for the Infinity Laplacian

Lipschitz regularity of almost minimizers in one-phase problems driven by the $p$-Laplace operator

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