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

2017/05/03 by Alejandro J. Castro, Salvador Rodríguez-López, Wolfgang Staubach · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.1090/tran/6958

Abstract

We establish the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Rⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.

Cited by

• A relation between the Dirichlet and the Regularity problem for parabolic equations
• The L^p regularity problem for parabolic operators with transversally independent coefficients
• Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis. [europepmc]
• On the Regularity Problem for Parabolic Operators and the Role of Half-Time Derivative. [europepmc]

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