L$^2$ well-posedness of boundary value problems for parabolic systems with measurable coefficients
| Autor: | Pascal Auscher, Kaj Nyström, Moritz Egert |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Uppsala], Uppsala University, Mathematical Research Science Institute, ANR-12-BS01-0013,HAB,Aux frontières de l'analyse Harmonique(2012), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2020 |
| Předmět: |
Dirichlet and Neumann problems
General Mathematics Mathematics::Analysis of PDEs [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] half-order derivative Dirac operator parabolic Kato square root estimate 01 natural sciences parabolic Dirac operator symbols.namesake Classical Analysis and ODEs (math.CA) FOS: Mathematics Order (group theory) (non-tangential) maximal functions Boundary value problem 0101 mathematics Mathematics 2010 MSC: Primary: 35K40 35K46 42B37. Secondary: 26A33 42B25 47A60 47D06 Applied Mathematics 010102 general mathematics Mathematical analysis Second order parabolic systems boundary layer operators First order Primary: 35K40 35K46 42B37. Secondary: 26A33 42B25 47A60 47D06 square function estimates 010101 applied mathematics Mathematics - Classical Analysis and ODEs a priori representations symbols Well posedness |
| Zdroj: | J. Eur. Math. Soc. (JEMS) 6 J. Eur. Math. Soc. (JEMS) 6, 2020, 22 (9), pp.2943--3058 Journal of the European Mathematical Society Journal of the European Mathematical Society, European Mathematical Society, 2020, 22 (9), pp.2943-3058. ⟨10.4171/JEMS/980⟩ |
| ISSN: | 1435-9855 1435-9863 |
| DOI: | 10.4171/jems/980 |
| Popis: | We prove the first positive results concerning boundary value problems in the upper half-space of second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green's representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in $L^2$-Sobolev spaces together with invertibility of layer potentials, and perturbation results. In the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversally independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation. Comment: Upload of the published version |
| Databáze: | OpenAIRE |
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