Hölder Continuity of the Minimizer of an Obstacle Problem with Generalized Orlicz Growth

Karppinen, Arttu; Lee, Mikyoung

doi:10.1093/imrn/rnab150

Cited by 13 publications

(7 citation statements)

References 45 publications

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“…Next step of proving regularity is to show that for bounded obstacles the solution is also bounded. The proof can be found in [38]. Proposition 4.4 ( [38]).…”

Section: Hölder Regularity Of the Obstacle Problemmentioning

confidence: 99%

Removable sets in elliptic equations with Musielak–Orlicz growth

Chlebicka

Karppinen

2021

Journal of Mathematical Analysis and Applications

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We characterize, in the terms of intrinsic Hausdorff measures, the size of removable sets for Hölder continuous solutions to elliptic equations with Musielak-Orlicz growth. In the general case we provide a result in the new scale that is more relevant and captures -as special cases -the classical results, slightly refines the ones provided for problems stated in the variable exponent and double phase spaces and essentially improves the known one in the Orlicz case.2010 Mathematics Subject Classification. 35J60, 35J70.

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“…Next step of proving regularity is to show that for bounded obstacles the solution is also bounded. The proof can be found in [38]. Proposition 4.4 ( [38]).…”

Section: Hölder Regularity Of the Obstacle Problemmentioning

confidence: 99%

Removable sets in elliptic equations with Musielak–Orlicz growth

Chlebicka

Karppinen

2021

Journal of Mathematical Analysis and Applications

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“…We note the following basic information on the existence, the uniqueness, and the Comparison Principle for the obstacle problem are provided in [37] and [16,Section 4]. Proposition 3.9 (Theorem 2, [16]).…”

Section: 2mentioning

confidence: 99%

“…The generalization of studies on removable sets for Hölder continuous solutions provided by [40] to the case of strongly non-uniformly elliptic operators has been carried out lately in [15,16]. There are available various regularity results for related quasiminimizers having Orlicz or generalized Orlicz growth [25,29,30,32,36,37,46]. For other recent developments in the understanding of the functional setting we refer also to [3,19,22,31].…”

Section: State Of Artmentioning

confidence: 99%

Generalized superharmonic functions with strongly nonlinear operator

Chlebicka¹,

Zatorska–Goldstein²

2020

Preprint

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We study properties of A-harmonic and A-superharmonic functions involving an operator having generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. In particular, Harnack's Principle and Minimum Principle are provided for A-superharmonic functions and boundary Harnack inequality is proven for A-harmonic functions.

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“…for some non-relabelled subsequence and generalized gradient 'D'. For well-posedness and basic properties of the obstacle problem see [22,Section 4] and [43]. By Lemma 3.16, the sequence {u j } is nondecreasing.…”

Section: 1mentioning

confidence: 99%

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

Chlebicka¹,

Giannetti²,

Zatorska–Goldstein³

2020

Preprint

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We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for A-superharmonic functions with nonlinear operator A : Ω × R n → R n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg-Wolff theorem on characterization of the dual of the Orlicz space.

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Hölder Continuity of the Minimizer of an Obstacle Problem with Generalized Orlicz Growth

Abstract: We prove local $C^{0,\alpha }$- and $C^{1,\alpha }$-regularity for the local solution to an obstacle problem with nonstandard growth. These results cover as special cases standard, variable exponent, double phase, and Orlicz growth.

Cited by 13 publications

References 45 publications

Removable sets in elliptic equations with Musielak–Orlicz growth

Removable sets in elliptic equations with Musielak–Orlicz growth

Generalized superharmonic functions with strongly nonlinear operator

Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

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