C1,α Regularity for Degenerate Fully Nonlinear Elliptic Equations with Neumann Boundary Conditions

Banerjee, Agnid; Verma, Ram Baran

doi:10.1007/s11118-021-09918-z

Cited by 4 publications

(10 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Differently from [42], we perform a simplest approach, which allows us to carry out the proof without using a change of variable scheme of viscosity solutions (a diffeomorphism for flat boundaries), which is a perfect fit for uniformly elliptic operators, but more technical for degenerate ones (cf. [8]). In fact, the strategy of proofs used in this session follows with corresponding adjustments to our unbalanced degeneracy setting, the ones from Birindelli-Demengel's works [10] and [12].…”

Section: Global Equicontinuity Of Solutions: Proof Of Theorem 12mentioning

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

View full text Add to dashboard Cite

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: Global Equicontinuity Of Solutions: Proof Of Theorem 12mentioning

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

View full text Add to dashboard Cite

show abstract

“…The existence of solution for analogous problems equipped with Dirichlet boundary conditions has been studied in [9], while the Neumann case is treated when α = 0 in [13], and for any α > −1 in [24]. On regularity results up to the boundary, which extend the regularity results of [16] , let us recall the results of [11] for the Dirichlet case, and those of [28], [3] for the Neumann problem.…”

Section: Introductionmentioning

confidence: 99%

Mixed boundary value problems for fully nonlinear degenerate or singular equations

Birindelli¹,

Demengel²,

Leoni³

2021

Preprint

View full text Add to dashboard Cite

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global Hölder estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global Hölder estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.

show abstract

“…Birindelli, Demengel and Leoni [14] proved the existence and uniqueness, and then a global Hölder regularity for the singular/degenerate case under the mixed boundary condition. Banerjee and Verma [6] case under the nonhomogeneous Neumann condition and on a C 2 domain. Ricarte [33] proved an optimal C 1,α regularity under a C 2 domain when F is convex.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand when |q| is large, the Lipschitz estimate can be proved by adapting Ishii-Lion's method in [20]. For the Neumann boundary case in [6], a boundary Hölder estimate on the flat domain was derived in the spirit of [23]. When |q| is small, a boundary Hölder estimate on the flat domain, for the uniformly elliptic equation that holds only where the gradient is large, was derived as in [24] by adapting the method of sliding cusps.…”

Section: Introductionmentioning

confidence: 99%

W2,-estimates for fully nonlinear elliptic equations with oblique boundary conditions

Byun

Han

2020

Journal of Differential Equations

View full text Add to dashboard Cite

C1,α Regularity for Degenerate Fully Nonlinear Elliptic Equations with Neumann Boundary Conditions

Cited by 4 publications

References 24 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

Mixed boundary value problems for fully nonlinear degenerate or singular equations

W2,-estimates for fully nonlinear elliptic equations with oblique boundary conditions

Contact Info

Product

Resources

About