2021/11/15 by Björn Augner, Augner, Björn
Mathematics · #35B65 #35D35 #35J58 #35K52 #42B15 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2111.07851
openalex publication_date 2021/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the theory of non-linear parabolic and elliptic partial differential\nequations, the notion of maximal regularity plays an essential role in\nestablishing existence, regularity and boundedness of solutions. There is a\nlong history of works where sufficient conditions for maximal regularity have\nbeen established: First scalar equations and systems of finitely many coupled\nequations have been considered. Around 2000, the vector-valued case with\ninfinite-dimensional range space E became accessible to the development and\nprogress in theory of \R-bounded operator families and its close\nconnection to the \H^\∞-calculus. The ground-braking results by\nDenk, Hieber and Pr "uss for \Lp-maximal regularity of vector-valued\nparabolic and elliptic boundary value problems, however, were restricted to\nboundary conditions with homogeneous principle parts of the boundary symbol, in\ncontrast to some previous results by Ladyszenskaya, Solonnikov and Uralceva for\nfinite-component systems which also allow for, e.g. both Dirichlet and (mixed)\nflux boundary conditions at the same position. In this manuscript we aim for\nclosing this gap, and extend the results of Denk, Hieber and Pr "uss to this\nslightly more general situation. To this end, we closely review the strategy\nused in their works and adapt it to the situation considered here.\n