𝐿²-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients

Castro, Alejandro J.; Rodríguez-López, Salvador; Staubach, Wolfgang

doi:10.1090/tran/6958

$L^2$-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients
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by Alejandro J. Castro, Salvador Rodríguez-López and Wolfgang Staubach PDF

Trans. Amer. Math. Soc. 370 (2018), 265-319 Request permission

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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