A degenerate fully nonlinear free transmission problem with variable exponents

Jesus, David

doi:10.1007/s00526-021-02144-w

Cited by 2 publications

(3 citation statements)

References 11 publications

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“…obtaining existence and interior 𝐶 1,𝛼 regularity of viscosity solutions. See also [31] for a degenerate fully nonlinear-free transmission problem, which addresses optimal pointwise estimates, where the degeneracy law varies in the domain. Consequently, after these breakthroughs and taking into account the previous highlights, according to our scientific knowledge, there is no investigation concerning existence/uniqueness, sharp, and global regularity for such class of problems (1.8) in a general scenario with unbalanced degeneracy and variable order.…”

Section: Introductionmentioning

confidence: 99%

“…\end{equation}$$We also recall the recent De Filippis' work [25], in which the author studied free transmission‐type problems of the form

\begin{equation*} {\left(|Du|^{p^+ {\chi }_{\lbrace u&gt;0\rbrace }+p_-{\chi }_{\lbrace u&lt;0\rbrace }}+a(x){\chi }_{\lbrace u&gt;0\rbrace }|Du|^q+b(x){\chi }_{\lbrace u&lt;0\rbrace }|Du|^s\right)}F(D^2 u)=f(x)\,\,\text{in }\,\,\Omega \end{equation*}

obtaining existence and interior

C^{1,\alpha }

regularity of viscosity solutions. See also [31] for a degenerate fully nonlinear‐free transmission problem, which addresses optimal pointwise estimates, where the degeneracy law varies in the domain.…”

Section: Introductionmentioning

confidence: 99%

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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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…\end{equation}$$We also recall the recent De Filippis' work [25], in which the author studied free transmission‐type problems of the form

\begin{equation*} {\left(|Du|^{p^+ {\chi }_{\lbrace u&gt;0\rbrace }+p_-{\chi }_{\lbrace u&lt;0\rbrace }}+a(x){\chi }_{\lbrace u&gt;0\rbrace }|Du|^q+b(x){\chi }_{\lbrace u&lt;0\rbrace }|Du|^s\right)}F(D^2 u)=f(x)\,\,\text{in }\,\,\Omega \end{equation*}

obtaining existence and interior

C^{1,\alpha }

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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show abstract

“…The estimates in [25] depend explicitly on θ 1 and θ 2 . See [24,30] for the analysis of interesting variants of the model in (3). The model in ( 3) is motivated by the study of fully nonlinear equations degenerating as a power of the gradient.…”

Section: Introductionmentioning

confidence: 99%

Existence of solutions to a fully nonlinear free transmission problem

Pimentel

Święch

2022

Journal of Differential Equations

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A degenerate fully nonlinear free transmission problem with variable exponents

Cited by 2 publications

References 11 publications

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

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

Existence of solutions to a fully nonlinear free transmission problem

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