Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities

Ho-Sik Lee; Youchan Kim; Ho-Sik Lee; Youchan Kim

doi:10.3934/cam.2025004

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 61-99. doi: 10.3934/cam.2025004

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Research article

Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities

Ho-Sik Lee ¹,
Youchan Kim ^{2
,
,}

1.
Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
2.
Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea

Received: 25 April 2024 Revised: 25 October 2024 Accepted: 25 October 2024 Published: 21 January 2025
Primary: 35B65; Secondary: 35K55, 46E30, 35R05

We obtain a boundary pointwise gradient estimate on a parabolic half cube $ Q_{2R} \cap \{ (x^{1}, x', t) \in \mathbb{R}^{n+1} : x^{1} > 0 \} $ for nonlinear parabolic equations with measurable nonlinearities, which are only assumed to be measurable in $ x^{1} $-variable. The estimates are obtained in terms of Riesz potential of the right-hand side measure and the oscillation of the boundary data, where the boundary data is given on $ Q_{2R} \cap \{ (x^{1}, x', t) \in \mathbb{R}^{n+1} : x^{1} = 0 \} $.
- nonlinear parabolic equations,
- Riesz potentials,
- measurable nonlinearities
Citation: Ho-Sik Lee, Youchan Kim. Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities[J]. Communications in Analysis and Mechanics, 2025, 17(1): 61-99. doi: 10.3934/cam.2025004

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Abstract

We obtain a boundary pointwise gradient estimate on a parabolic half cube $ Q_{2R} \cap \{ (x^{1}, x', t) \in \mathbb{R}^{n+1} : x^{1} > 0 \} $ for nonlinear parabolic equations with measurable nonlinearities, which are only assumed to be measurable in $ x^{1} $-variable. The estimates are obtained in terms of Riesz potential of the right-hand side measure and the oscillation of the boundary data, where the boundary data is given on $ Q_{2R} \cap \{ (x^{1}, x', t) \in \mathbb{R}^{n+1} : x^{1} = 0 \} $.

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