Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

Marco Cirant; Gianmaria Verzini

doi:10.1016/j.aim.2022.108700

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

Marco Cirant, Gianmaria Verzini

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

10 Scopus citations

Abstract

We study the local Hölder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When [Formula presented] (N is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent [Formula presented]. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.

Original language	English (US)
Pages (from-to)	108700
Journal	Advances in Mathematics
Volume	409
DOIs	https://doi.org/10.1016/j.aim.2022.108700
State	Published - Sep 21 2022
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-10-07
Acknowledged KAUST grant number(s): CRG2021-4674
Acknowledgements: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Work partially supported by the project Vain-Hopes within the program VALERE-Università degli Studi della Campania “Luigi Vanvitelli”, by the Portuguese government through FCT/Portugal under the project PTDC/MAT-PUR/1788/2020, and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field Games: models, theory, and computational aspects”.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2022.108700

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title = "Local H{\"o}lder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms",

abstract = "We study the local H{\"o}lder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When [Formula presented] (N is the dimension of the Euclidean space), we obtain the optimal H{\"o}lder continuity exponent [Formula presented]. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.",

author = "Marco Cirant and Gianmaria Verzini",

note = "KAUST Repository Item: Exported on 2022-10-07 Acknowledged KAUST grant number(s): CRG2021-4674 Acknowledgements: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit{\`a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Work partially supported by the project Vain-Hopes within the program VALERE-Universit{\`a} degli Studi della Campania “Luigi Vanvitelli”, by the Portuguese government through FCT/Portugal under the project PTDC/MAT-PUR/1788/2020, and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field Games: models, theory, and computational aspects”. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2022",

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doi = "10.1016/j.aim.2022.108700",

language = "English (US)",

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journal = "Advances in Mathematics",

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AU - Cirant, Marco

AU - Verzini, Gianmaria

N1 - KAUST Repository Item: Exported on 2022-10-07 Acknowledged KAUST grant number(s): CRG2021-4674 Acknowledgements: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Work partially supported by the project Vain-Hopes within the program VALERE-Università degli Studi della Campania “Luigi Vanvitelli”, by the Portuguese government through FCT/Portugal under the project PTDC/MAT-PUR/1788/2020, and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field Games: models, theory, and computational aspects”. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2022/9/21

Y1 - 2022/9/21

N2 - We study the local Hölder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When [Formula presented] (N is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent [Formula presented]. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.

AB - We study the local Hölder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When [Formula presented] (N is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent [Formula presented]. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.

UR - http://hdl.handle.net/10754/682261

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DO - 10.1016/j.aim.2022.108700

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JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

Abstract

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