Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications

Silva, João Vítor da; Ricarte, Gleydson C.

doi:10.1007/s00526-020-01820-7

Cited by 15 publications

(19 citation statements)

References 55 publications

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“…It is noteworthy to mention that De Filippis in [17] introduced the double phase type degeneracies to the fully nonlinear equation

[{false| D u false|}^{p} + a (x) {false| D u false|}^{q}] F false(D^{2} u false) = f false(x false), 0 < p ⩽ q,

and proved the

C^{1, γ}

local regularity for viscosity solutions. Moreover, the sharp local

C^{1, γ}

geometric regularity estimates for bounded viscosity solutions were obtained by da Silva and Ricarte in [27]. Meanwhile, under rather general conditions, Bronzi et al .…”

Section: Introductionmentioning

confidence: 94%

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Fang

Rădulescu

Zhang

2021

Bulletin of London Math Soc

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“…It is noteworthy to mention that De Filippis in [17] introduced the double phase type degeneracies to the fully nonlinear equation

[{false| D u false|}^{p} + a (x) {false| D u false|}^{q}] F false(D^{2} u false) = f false(x false), 0 < p ⩽ q,

and proved the

C^{1, γ}

local regularity for viscosity solutions. Moreover, the sharp local

C^{1, γ}

geometric regularity estimates for bounded viscosity solutions were obtained by da Silva and Ricarte in [27]. Meanwhile, under rather general conditions, Bronzi et al .…”

Section: Introductionmentioning

confidence: 94%

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Fang

Rădulescu

Zhang

2021

Bulletin of London Math Soc

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“…and proved the C 1,γ local regularity for viscosity solutions. Moreover, the sharp local C 1,γ geometric regularity estimates for bounded viscosity solutions were obtained by da Silva and Ricarte in [26]. Meanwhile, under rather general conditions, Bronzi, Pimentel, Rampasso and Teixeira in [8] proved that viscosity solutions to the following variable exponent fully nonlinear elliptic equations |Du| p(x) F (D 2 u) = f are locally of class C 1,γ for a universal constant γ ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 91%

“…In this paper, motivated by the results in [8,17,26] we consider the fully nonlinear elliptic equations with variable-exponent nonhomogeneous degeneracy of the form (1.1). By making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques, we show that the viscosity solutions to (1.1) are locally of class C 1,α (Ω).…”

Section: Introductionmentioning

confidence: 99%

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Fang¹,

Rădulescu²,

Zhang³

2021

Preprint

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“…Furthermore, the degeneracy character of the operator naturally leads to split the proof into two steps, according to whether the gradient is "large or small". Before that, we recall that the interior estimates from [daSR20] and [DeF20] and the fact that…”

Section: A Harnack Type Inequalitymentioning

confidence: 99%

Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

Silva¹,

Rampasso²,

Ricarte³

et al. 2021

Preprint

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We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such non-linear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli's classification scheme: Flat and Lipschitz free boundaries are locally C 1,β for some 0 < β (universal) < 1. Particularly, our findings are new even for the toy modelWe also bring to light other interesting doubly degenerate settings where our results still work. Finally, we present some key tools in the theory of degenerate fully nonlinear PDEs, which may have their own mathematical importance and applicability.

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Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications

Cited by 15 publications

References 55 publications

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

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