Green's functions for parabolic systems of second order in time-varying domains

Hongjie Dong; Seick Kim

doi:10.3934/cpaa.2014.13.1407

Article Contents

2014, Volume 13, Issue 4: 1407-1433. Doi: 10.3934/cpaa.2014.13.1407

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Green's functions for parabolic systems of second order in time-varying domains

Hongjie Dong^1, and
Seick Kim^2,

1.
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912
2.
Department of Mathematics, Yonsei University, Seoul 120-749

Received: February 28, 2013

Revised: December 31, 2012

Published: June 2015

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Abstract

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and $1/2$-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate. In particular, our results apply to complex perturbations of a single real equation.

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Mathematics Subject Classification: Primary: 35A08, 35K40; Secondary: 35B45.

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