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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Sharp estimates for the gradient of solutions to the heat equation

Authors: G. Kresin and V. Maz′ya
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 495-507
MSC (2010): Primary 35K05; Secondary 26D20
Published electronically: April 30, 2020
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Abstract: Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space $ L^p$. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.

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Additional Information

G. Kresin
Affiliation: Department of Mathematics, Ariel University, Ariel 40700, Israel

V. Maz′ya
Affiliation: Department of Mathematical Sciences, University of Liverpool, M&O Building, Liverpool, L69 3BX, UK; Department of Mathematics, Linköping University, SE-58183 Linköping, Sweden; RUDN University, 6 Miklukho-Maklay St., 117198 Moscow, Russia

Keywords: Heat equation, sharp pointwise estimates for the gradient, first and second boundary value problems
Received by editor(s): June 6, 2018
Published electronically: April 30, 2020
Dedicated: In memory of Solomon G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society