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On a new type of boundary condition

Authors
  • Pedregal, Pablo1
  • 1 INEI, Universidad de Castilla-La Mancha, ETSI Industriales, Ciudad Real, 13071, Spain , Ciudad Real (Spain)
Type
Published Article
Journal
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
Publisher
Springer International Publishing
Publication Date
Nov 27, 2021
Volume
116
Issue
1
Identifiers
DOI: 10.1007/s13398-021-01189-y
Source
Springer Nature
Keywords
Disciplines
  • Original Paper
License
Yellow

Abstract

Pushed by inverse problems in conductivity in the 3-dimensional setting, we introduce new types of boundary conditions for variational and PDE problems, that in some sense cover the middle space between the classical Dirichlet and Neumann conditions, meant in a essentially different way with respect to mixed boundary conditions. These new boundary conditions are associated with special subspaces of Sobolev spaces between H01(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1_0(\Omega )$$\end{document} and the full space H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega )$$\end{document}. Though problems can be considered in W1,p(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1, p}(\Omega )$$\end{document} for p≠2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ne 2$$\end{document}, in this initial contribution we just examine existence and optimality for regular variational problems under typical assumptions within the scope of H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\Omega )$$\end{document}. In addition to the existence of minimizers, we would like to stress the intriguing form of optimality at the boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}. We especially treat the case N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document}, which is the most interesting case, and describe similar conditions in any dimension N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}. The numerical approximation definitely requires new ideas.

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