The Wolff potential estimate for solutions to elliptic equations with signed data

Hara, Takanobu

doi:10.1007/s00229-015-0805-z

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…for some c = c(data, γ) > 0. Analogically, by (50) for ū u u j and with δ = 2 −j , by letting j → ∞, Fatou's lemma on the left-hand side of the resultant inequality and (78) on its right-hand side, we get…”

Section: Measure Data A-harmonic Approximationmentioning

confidence: 97%

Wolff potentials and measure data vectorial problems with Orlicz growth

Chlebicka¹,

Youn²,

Zatorska–Goldstein³

2021

Preprint

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We study solutions to measure data elliptic systems with Uhlenbeck-type structure that involve operator of divergence form, depending continuously on the spacial variable, and exposing doubling Orlicz growth with respect to the second variable. Pointwise estimates for the solutions that we provide are expressed in terms of a nonlinear potential of generalized Wolff type. Not only we retrieve the recent sharp results proven for p-Laplace systems, but additionally our study covers the natural scope of operators with similar structure and natural class of Orlicz growth.

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Section: Measure Data A-harmonic Approximationmentioning

confidence: 97%

Wolff potentials and measure data vectorial problems with Orlicz growth

Chlebicka¹,

Youn²,

Zatorska–Goldstein³

2021

Preprint

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“…including weighted and vectorial ones, see Kilpeläinen-Malý [37], [38], Mikkonen [49], Hara [24], the monograph Heinonen-Kilpeläinen-Martio [26,Theorem 21.21] and the expository papers Kuusi-Mingione [44], [45]. On metric spaces, such estimates have been obtained for Cheeger p-harmonic functions in Björn-MacManus-Shanmugalingam [14] and Hara [25].…”

Section: Elliptic Equations In Divergence Formmentioning

confidence: 99%

Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions

Björn¹,

Björn²,

Lehrbäck³

2021

Preprint

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In a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, we prove sharp growth and integrability results for p-harmonic Green functions and their minimal p-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R n and on manifolds.The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

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“…In fact, for the method of our reasoning the most influential was later proof provided by Korte and Kuusi [48] based on the ideas by Trudinger and Wang [69]. Similar scheme was used also in [38,52]. This approach requires a basic toolkit of potential analysis provided in [24].…”

Section: Introductionmentioning

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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The Wolff potential estimate for solutions to elliptic equations with signed data

Cited by 7 publications

References 17 publications

Wolff potentials and measure data vectorial problems with Orlicz growth

Wolff potentials and measure data vectorial problems with Orlicz growth

Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions

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

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