Nonlinear obstacle problems with double phase in the borderline case

Byun, Sun‐Sig; Cho, Yumi; Oh, Jehan

doi:10.1002/mana.201800277

Cited by 5 publications

(1 citation statement)

References 55 publications

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“…These include improved integrability and differentiability results in the case of variable exponent functionals ´Ω |Du| p(x) dx [24,38,25,35], as well as the double phase functional ´Ω |Du| p + a(x)|Du| q dx [11,16,67]. Another direction of research has been Caldéron-Zygmund estimates for both double phase [10] and variable exponent [54] obstacle problems. We mention that improved integrability results are also available in the setting of almost linear growth [39,59] as well as certain parabolic settings, see for example [6,26] for results and further references.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Global higher integrability for minimisers of convex functionals with (p,q)-growth

Koch

2021

Calc. Var.

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We prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

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Section: Introduction and Resultsmentioning

confidence: 99%

Global higher integrability for minimisers of convex functionals with (p,q)-growth

Koch

2021

Calc. Var.

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Global higher integrability for minimisers of convex obstacle problems with (p,q)-growth

Koch

2022

Calc. Var.

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We prove global $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) -regularity for minimisers of $${{\mathscr {F}}(u)=\int _\Omega F(x,Du)\,{\mathrm{d}}x}$$ F ( u ) = ∫ Ω F ( x , D u ) d x satisfying $$u\ge \psi $$ u ≥ ψ for a given Sobolev obstacle $$\psi $$ ψ . $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and natural (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n . In the autonomous case $$F\equiv F(z)$$ F ≡ F ( z ) we can improve the gap to $$q<\min \left( \frac{np}{n-1},p+1\right) $$ q < min np n - 1 , p + 1 , a result new even in the unconstrained case.

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Recent developments in problems with nonstandard growth and nonuniform ellipticity

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Rădulescu

2021

Journal of Mathematical Analysis and Applications

206

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Nonlinear obstacle problems with double phase in the borderline case

Cited by 5 publications

References 55 publications

Global higher integrability for minimisers of convex functionals with (p,q)-growth

Global higher integrability for minimisers of convex functionals with (p,q)-growth

Global higher integrability for minimisers of convex obstacle problems with (p,q)-growth

Recent developments in problems with nonstandard growth and nonuniform ellipticity

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