Fully nonlinear Hamilton–Jacobi equations of degenerate type

David Jesus; Edgard A. Pimentel; José Miguel Urbano

doi:10.1016/j.na.2022.113181

Fully nonlinear Hamilton–Jacobi equations of degenerate type

David Jesus, Edgard A. Pimentel, José Miguel Urbano^*

^*Corresponding author for this work

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

1 Scopus citations

Abstract

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

Original language	English (US)
Article number	113181
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	227
DOIs	https://doi.org/10.1016/j.na.2022.113181
State	Published - Feb 2023

Bibliographical note

Funding Information:
EP partially supported by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ) and by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021 ).

Funding Information:
JMU partially supported by the King Abdullah University of Science and Technology (KAUST), Saudi Arabia , by Fundação para a Ciência e a Tecnologia , through project PTDC/MAT-PUR/28686/2017 , and by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ).

Funding Information:
DJ was supported by Fundação para a Ciência e a Tecnologia, Portugal , through scholarship PD/BD/150354/2019 , under POCH funds, co-financed by the European Social Fund and Portuguese National Funds from MCTES, and by the Centre for Mathematics of the University of Coimbra, Portugal ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ).

Publisher Copyright:
© 2022 Elsevier Ltd

Keywords

Degenerate elliptic operators
Hamilton–Jacobi equation
Lipschitz regularity
Two-phase free boundary problems

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2022.113181

Cite this

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title = "Fully nonlinear Hamilton–Jacobi equations of degenerate type",

abstract = "We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.",

keywords = "Degenerate elliptic operators, Hamilton–Jacobi equation, Lipschitz regularity, Two-phase free boundary problems",

author = "David Jesus and Pimentel, {Edgard A.} and Urbano, {Jos{\'e} Miguel}",

note = "Funding Information: EP partially supported by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ) and by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021 ). Funding Information: JMU partially supported by the King Abdullah University of Science and Technology (KAUST), Saudi Arabia , by Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia , through project PTDC/MAT-PUR/28686/2017 , and by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ). Funding Information: DJ was supported by Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia, Portugal , through scholarship PD/BD/150354/2019 , under POCH funds, co-financed by the European Social Fund and Portuguese National Funds from MCTES, and by the Centre for Mathematics of the University of Coimbra, Portugal ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ). Publisher Copyright: {\textcopyright} 2022 Elsevier Ltd",

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AU - Pimentel, Edgard A.

AU - Urbano, José Miguel

N1 - Funding Information: EP partially supported by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ) and by FAPERJ (grants E26/200.002/2018 and E26/201.390/2021 ). Funding Information: JMU partially supported by the King Abdullah University of Science and Technology (KAUST), Saudi Arabia , by Fundação para a Ciência e a Tecnologia , through project PTDC/MAT-PUR/28686/2017 , and by the Centre for Mathematics of the University of Coimbra ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ). Funding Information: DJ was supported by Fundação para a Ciência e a Tecnologia, Portugal , through scholarship PD/BD/150354/2019 , under POCH funds, co-financed by the European Social Fund and Portuguese National Funds from MCTES, and by the Centre for Mathematics of the University of Coimbra, Portugal ( UIDB/00324/2020 , funded by the Portuguese Government through FCT/MCTES ). Publisher Copyright: © 2022 Elsevier Ltd

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N2 - We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

AB - We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

KW - Degenerate elliptic operators

KW - Hamilton–Jacobi equation

KW - Lipschitz regularity

KW - Two-phase free boundary problems

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M1 - 113181

ER -

Fully nonlinear Hamilton–Jacobi equations of degenerate type

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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