Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries

Silva, João Vítor da; Júnior, Raimundo Alves Leitão; Ricarte, Gleydson C.

doi:10.1002/mana.201800555

Cited by 8 publications

(5 citation statements)

References 35 publications

(65 reference statements)

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“…The case m ∈ (0, 1) has been studied in many contexts. For instance, in [16] and [17], the authors treat the case m ∈ (0, 1) as one-phase free boundary problems. An advantage of the techniques used here is the gain of smoothness along subsets of the zero level set even for signchanging solutions.…”

Section: Assumptions Main Results and Further Insightsmentioning

confidence: 99%

Escobar type theorems for elliptic fully nonlinear degenerate equations

Abanto¹,

Espinar²

2019

American Journal of Mathematics

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In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are the hemisphere (or a geodesic ball in S m ), punctured balls and annular domains.Our results extend those of Escobar in [8] when m ≥ 3, and Hang-Wang in [18] and Jimenez in [19] when m = 2.

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Section: Assumptions Main Results and Further Insightsmentioning

confidence: 99%

Escobar type theorems for elliptic fully nonlinear degenerate equations

Abanto¹,

Espinar²

2019

American Journal of Mathematics

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“…In brief, our strategy follows the seminal ideas and successful programme directly by Teixeira and Urbano's works in the past decade [40,42,44] (see also [2,4,9,17] for similar developments-see also Teixeira's recent paper [41] for an excellent reference for these set of pivotal insights). Nevertheless, we establish a finer and sharper version of [4, lemma 3.2], which allows us to transfer (in a continuous fashion) available regularity estimates for the homogeneous case to the inhomogeneous one, under a suitable smallness regime on the data.…”

Section: Model Equationmentioning

confidence: 99%

“…Furthermore, the presence of such degeneracy law suggests the use of intrinsic scaling and geometric tangential techniques adjusted to our context. For this reason, it is necessary to consider several new aspects in the original argument presented, for instance, as see [2,9,42] in the scenario of evolutionary p-Laplacian type equations, and [4] for the corresponding doubly degenerate model.…”

Section: Introductionmentioning

confidence: 99%

Geometric estimates for doubly nonlinear parabolic PDEs

2022

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In this manuscript, we establish C loc α , α θ regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂ u ∂ t − d i v ( m | u | m − 1 | ∇ u | p − 2 ∇ u ) = f ( x , t ) in Ω T ≔ Ω × ( 0 , T ) , where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.

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“…Aftermath, da Silva-Ricarte in [21] improved De Filippis' result and addressed a variety of applications in nonlinear elliptic models and related free boundary problems. Concerning fully nonlinear models of (single) degenerate type (i.e., 𝔞 ≡ 0 in (1.7)) the list of contributions is fairly diverse, including aspects such as existence/uniqueness issues, Harnack inequality, Alexandroff-Bakelman-Pucci (ABP) estimates [20] and [9], Liouville-type results [19], local Hölder and Lipschitz estimates, local gradient estimates [3,[10][11][12], and [30], as well as connections with a variety of free boundary problems of Bernoulli type [20], obstacle type [23], singular perturbation type [4,9] and [37], and dead-core type [19], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

Silva

Júnior

Rampasso

et al. 2023

Journal of London Math Soc

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We establish the existence and sharp global regularity results (C0,γ$C^{0, \gamma }$, C0,1$C^{0, 1}$ and C1,α$C^{1, \alpha }$ estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function frakturafalse(·false)⩾0$\mathfrak {a}(\cdot )\geqslant 0$. The model case in question is given by -0.16em-0.16em{false|Dufalse|pfalse(xfalse)+fraktura(x)false|Dufalse|qfalse(xfalse)scriptMλ,Λ+false(D2ufalse)=ffalse(xfalse)inΩu(x)=g(x)on∂Ω,$$\begin{equation*} \!\!{\left\lbrace \! \def\eqcellsep{&}\begin{array}{rcl} {\left[|Du|^{p(x)}+\mathfrak {a}(x)|Du|^{q(x)}\right]}\mathcal {M}_{\lambda , \Lambda }^{+}(D^2 u) = f(x) & \!\text{in} & \!\Omega \\[3pt] u(x) = g(x) & \!\text{on} & \!\partial \Omega , \end{array} \right.} \end{equation*}$$for a bounded, regular, and open set normalΩ⊂Rn$\Omega \subset \mathbb {R}^n$, and appropriate continuous data pfalse(·false),qfalse(·false)$p(\cdot ), q(\cdot )$, ffalse(·false)$f(\cdot )$, and gfalse(·false)$g(\cdot )$. Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

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Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries

Cited by 8 publications

References 35 publications

Escobar type theorems for elliptic fully nonlinear degenerate equations

Escobar type theorems for elliptic fully nonlinear degenerate equations

Geometric estimates for doubly nonlinear parabolic PDEs

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

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