Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications

Shigeaki Koike; Andrzej Święch; Shota Tateyama

doi:10.1016/j.na.2019.03.005

Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications

Shigeaki Koike, Andrzej Święch, Shota Tateyama

School of Advanced Science and Engineering

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

1 Citation (Scopus)

Abstract

The weak Harnack inequality for L ^p -viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of L ^p -viscosity solutions is derived from the weak Harnack inequality for L ^p -viscosity supersolutions. The local maximum principle for L ^p -viscosity subsolutions and the Harnack inequality for L ^p -viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.

Original language	English
Pages (from-to)	264-289
Number of pages	26
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	185
DOIs	https://doi.org/10.1016/j.na.2019.03.005
Publication status	Published - 2019 Aug

Keywords

Fully nonlinear parabolic equations
Harnack inequality
Maximum principle
Viscosity solutions

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2019.03.005

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title = "Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications",

abstract = " The weak Harnack inequality for L p -viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that H{\"o}lder continuity of L p -viscosity solutions is derived from the weak Harnack inequality for L p -viscosity supersolutions. The local maximum principle for L p -viscosity subsolutions and the Harnack inequality for L p -viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives. ",

keywords = "Fully nonlinear parabolic equations, Harnack inequality, Maximum principle, Viscosity solutions",

author = "Shigeaki Koike and Andrzej {\'S}wi{\c e}ch and Shota Tateyama",

note = "Funding Information: S. Koike is supported in part by Grant-in-Aid for Scientific Research (Nos. 16H06339 , 16H03948 , 16H03946 ) of Japan Society for the Promotion of Science . S. Tateyama is supported by Grant-in-Aid for Japan Society for the Promotion of Science Research Fellow 16J02399 . ",

year = "2019",

month = aug,

doi = "10.1016/j.na.2019.03.005",

language = "English",

volume = "185",

pages = "264--289",

journal = "Nonlinear Analysis, Theory, Methods and Applications",

issn = "0362-546X",

publisher = "Elsevier Limited",

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TY - JOUR

T1 - Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications

AU - Koike, Shigeaki

AU - Święch, Andrzej

AU - Tateyama, Shota

N1 - Funding Information: S. Koike is supported in part by Grant-in-Aid for Scientific Research (Nos. 16H06339 , 16H03948 , 16H03946 ) of Japan Society for the Promotion of Science . S. Tateyama is supported by Grant-in-Aid for Japan Society for the Promotion of Science Research Fellow 16J02399 .

PY - 2019/8

Y1 - 2019/8

N2 - The weak Harnack inequality for L p -viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of L p -viscosity solutions is derived from the weak Harnack inequality for L p -viscosity supersolutions. The local maximum principle for L p -viscosity subsolutions and the Harnack inequality for L p -viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.

AB - The weak Harnack inequality for L p -viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of L p -viscosity solutions is derived from the weak Harnack inequality for L p -viscosity supersolutions. The local maximum principle for L p -viscosity subsolutions and the Harnack inequality for L p -viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.

KW - Fully nonlinear parabolic equations

KW - Harnack inequality

KW - Maximum principle

KW - Viscosity solutions

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JO - Nonlinear Analysis, Theory, Methods and Applications

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SN - 0362-546X

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