Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

Antonella Nastasi; Cintia Pacchiano Camacho

doi:10.1007/s00526-021-02099-y

Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

Antonella Nastasi^*
, Cintia Pacchiano Camacho

^*Corresponding author for this work

Department of Mathematics and Systems Analysis

University of Palermo

Research output: Contribution to journal › Article › Scientific › peer-review

6 Citations (Scopus)

Abstract

Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.

Original language	English
Article number	227
Number of pages	37
Journal	Calculus of Variations and Partial Differential Equations
Volume	60
Issue number	6
DOIs	https://doi.org/10.1007/s00526-021-02099-y
Publication status	Published - Dec 2021
MoE publication type	A1 Journal article-refereed

Funding

The authors wish to thank Professor Juha Kinnunen for supporting us in our research and for all the enlightening discussions and advice. Special thanks go to Professor Paolo Marcellini for the panoramical view he gently provided on the existing literature and open questions on the subject. The authors are grateful to the referees for their careful reading and the useful comments. The second author was supported by a doctoral training Grant for 2021 from the Väisälä Fund.

Keywords

(p, q)-Laplace operator
Measure metric spaces
Minimal p-weak upper gradient
Minimizer

Access to Document

10.1007/s00526-021-02099-y

https://arxiv.org/abs/2104.06436

Cite this

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title = "Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral",

abstract = "Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincar{\'e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H{\"o}lder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville{\textquoteright}s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H{\"o}lder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.",

keywords = "(p, q)-Laplace operator, Measure metric spaces, Minimal p-weak upper gradient, Minimizer",

author = "Antonella Nastasi and \{Pacchiano Camacho\}, Cintia",

note = "Funding Information: The authors wish to thank Professor Juha Kinnunen for supporting us in our research and for all the enlightening discussions and advice. Special thanks go to Professor Paolo Marcellini for the panoramical view he gently provided on the existing literature and open questions on the subject. The authors are grateful to the referees for their careful reading and the useful comments. The second author was supported by a doctoral training Grant for 2021 from the V{\"a}is{\"a}l{\"a} Fund. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

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N2 - Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.

AB - Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.

KW - (p, q)-Laplace operator

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KW - Minimal p-weak upper gradient

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