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$ L^2$-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients


Authors: Alejandro J. Castro, Salvador Rodríguez-López and Wolfgang Staubach
Journal: Trans. Amer. Math. Soc. 370 (2018), 265-319
MSC (2010): Primary 35K20, 42B20
DOI: https://doi.org/10.1090/tran/6958
Published electronically: June 27, 2017
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Abstract: We establish the $ L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with time-independent Hölder-continuous diffusion coefficients on bounded Lipschitz domains in $ \mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer potentials, which is in turn based on Fredholm theory and a systematic transference scheme which yields suitable parabolic Rellich-type estimates.


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Alejandro J. Castro
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Address at time of publication: Department of Mathematics, Nazarbayev University, 010000 Astana, Kazakhstan
Email: alejandro.castilla@nu.edu.kz

Salvador Rodríguez-López
Affiliation: Department of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden
Email: s.rodriguez-lopez@math.su.se

Wolfgang Staubach
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: wolfgang.staubach@math.uu.se

DOI: https://doi.org/10.1090/tran/6958
Keywords: Boundary value problems, parabolic equations, Lipschitz domains, layer potentials, Rellich estimates
Received by editor(s): March 18, 2016
Published electronically: June 27, 2017
Additional Notes: The first author was partially supported by Swedish Research Council Grant 621-2011-3629
The second author was partially supported by Spanish Government grant MTM2013-40985-P
The third author was partially supported by a grant from the Crafoord Foundation
Article copyright: © Copyright 2017 American Mathematical Society