Skip to main content

2016 | OriginalPaper | Buchkapitel

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

verfasst von : Robert Denk, Tim Seger

Erschienen in: Recent Developments of Mathematical Fluid Mechanics

Verlag: Springer Basel

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals \(L^{p}(\Omega )\). However, much less is known for the \(W_{p}^{s}(\Omega )\)-realization, s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right-hand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962)MathSciNetCrossRefMATH S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119–147 (1962)MathSciNetCrossRefMATH
2.
Zurück zum Zitat M. Agranovich, R. Denk, M. Faierman, Weakly smooth nonselfadjoint spectral elliptic boundary problems, in Spectral Theory, Microlocal Analysis, Singular Manifolds, Math. Top., vol. 14 (Akademie Verlag, Berlin, 1997), pp. 138–199 M. Agranovich, R. Denk, M. Faierman, Weakly smooth nonselfadjoint spectral elliptic boundary problems, in Spectral Theory, Microlocal Analysis, Singular Manifolds, Math. Top., vol. 14 (Akademie Verlag, Berlin, 1997), pp. 138–199
3.
Zurück zum Zitat M.S. Agranovich, Elliptic boundary problems, in Partial Differential Equations, IX, Encyclopaedia Mathematical Science, vol. 79 (Springer, Berlin, 1997), pp. 1–144, 275–281. [Translated from the Russian by the author] M.S. Agranovich, Elliptic boundary problems, in Partial Differential Equations, IX, Encyclopaedia Mathematical Science, vol. 79 (Springer, Berlin, 1997), pp. 1–144, 275–281. [Translated from the Russian by the author]
4.
Zurück zum Zitat M.S. Agranovich, M.I. Vishik, Elliptic problems with a parameter and parabolic systems of general form. Russ. Math. Surv. 19, 53–157 (1964)CrossRefMATH M.S. Agranovich, M.I. Vishik, Elliptic problems with a parameter and parabolic systems of general form. Russ. Math. Surv. 19, 53–157 (1964)CrossRefMATH
5.
Zurück zum Zitat H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract linear theory, Monographs in Mathematics, vol. 89 (Birkhäuser, Boston, 1995) H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract linear theory, Monographs in Mathematics, vol. 89 (Birkhäuser, Boston, 1995)
6.
Zurück zum Zitat H. Amann, Anisotropic function spaces and maximal regularity for parabolic problems. Part 1: Function spaces, Jindr̆ich Nec̆as Center for Mathematical Modeling Lecture Notes, vol. 6 (Matfyzpress, Prague, 2009) H. Amann, Anisotropic function spaces and maximal regularity for parabolic problems. Part 1: Function spaces, Jindr̆ich Nec̆as Center for Mathematical Modeling Lecture Notes, vol. 6 (Matfyzpress, Prague, 2009)
8.
Zurück zum Zitat R. Denk, M. Dreher, Resolvent estimates for elliptic systems in function spaces of higher regularity. Electron. J. Differ. Equ. 109, 12 (2011)MathSciNetMATH R. Denk, M. Dreher, Resolvent estimates for elliptic systems in function spaces of higher regularity. Electron. J. Differ. Equ. 109, 12 (2011)MathSciNetMATH
9.
Zurück zum Zitat R. Denk, M. Faierman, M. Möller, An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108(3), 289–317 (2002)MathSciNetCrossRefMATH R. Denk, M. Faierman, M. Möller, An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108(3), 289–317 (2002)MathSciNetCrossRefMATH
10.
Zurück zum Zitat R. Denk, M. Hieber, J. Prüss, \(\mathcal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003) R. Denk, M. Hieber, J. Prüss, \(\mathcal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003)
11.
Zurück zum Zitat R. Denk, J. Saal, J. Seiler, Bounded \(H_{\infty }\)-calculus for pseudo-differential Douglis-Nirenberg systems of mild regularity. Math. Nachr. 282(3), 386–407 (2009)MathSciNetCrossRefMATH R. Denk, J. Saal, J. Seiler, Bounded \(H_{\infty }\)-calculus for pseudo-differential Douglis-Nirenberg systems of mild regularity. Math. Nachr. 282(3), 386–407 (2009)MathSciNetCrossRefMATH
12.
Zurück zum Zitat M. Faierman, On the resolvent arising in a parameter-elliptic multi-order boundary problem. Math. Nachr. 285(13), 1643–1670 (2012)MathSciNetCrossRefMATH M. Faierman, On the resolvent arising in a parameter-elliptic multi-order boundary problem. Math. Nachr. 285(13), 1643–1670 (2012)MathSciNetCrossRefMATH
13.
Zurück zum Zitat G. Geymonat, P. Grisvard, Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici. Rendiconti del Seminario Matematico della Universit di Padova 38, 121–173 (1967)MathSciNetMATH G. Geymonat, P. Grisvard, Alcuni risultati di teoria spettrale per i problemi ai limiti lineari ellittici. Rendiconti del Seminario Matematico della Universit di Padova 38, 121–173 (1967)MathSciNetMATH
14.
Zurück zum Zitat P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24 (Pitman Advanced Publishing Program, Boston, MA, 1985)MATH P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24 (Pitman Advanced Publishing Program, Boston, MA, 1985)MATH
15.
Zurück zum Zitat G. Grubb, N.J. Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in L p Sobolev spaces. Acta Math. 171(2), 165–229 (1993)MathSciNetCrossRefMATH G. Grubb, N.J. Kokholm, A global calculus of parameter-dependent pseudodifferential boundary problems in L p Sobolev spaces. Acta Math. 171(2), 165–229 (1993)MathSciNetCrossRefMATH
16.
Zurück zum Zitat D. Guidetti, A maximal regularity result with applications to parabolic problems with nonhomogeneous boundary conditions. Rendiconti del Seminario Matematico della Universit di Padova 84, 1–37 (1990)MathSciNetMATH D. Guidetti, A maximal regularity result with applications to parabolic problems with nonhomogeneous boundary conditions. Rendiconti del Seminario Matematico della Universit di Padova 84, 1–37 (1990)MathSciNetMATH
18.
Zurück zum Zitat D. Guidetti, On boundary value problems for parabolic equations of higher order in time. J. Differ. Equ. 124(1), 1–26 (1996)MathSciNetCrossRefMATH D. Guidetti, On boundary value problems for parabolic equations of higher order in time. J. Differ. Equ. 124(1), 1–26 (1996)MathSciNetCrossRefMATH
19.
Zurück zum Zitat M. Nesensohn, Randwertprobleme in Sobolevräumen höherer Ordnung. Diploma thesis, University of Konstanz, 2009 M. Nesensohn, Randwertprobleme in Sobolevräumen höherer Ordnung. Diploma thesis, University of Konstanz, 2009
20.
Zurück zum Zitat J.A. Roitberg, Z.G. Sheftel, Boundary value problems with a parameter for systems elliptic in the sense of Douglis-Nirenberg. Ukrain. Mat. Ž. 19(1), 115–120 (1967)MathSciNet J.A. Roitberg, Z.G. Sheftel, Boundary value problems with a parameter for systems elliptic in the sense of Douglis-Nirenberg. Ukrain. Mat. Ž. 19(1), 115–120 (1967)MathSciNet
21.
Zurück zum Zitat T. Seger, Elliptic-parabolic systems with applications to lithium-ion battery models. Ph.D. thesis, University of Konstanz, 2013 T. Seger, Elliptic-parabolic systems with applications to lithium-ion battery models. Ph.D. thesis, University of Konstanz, 2013
22.
Zurück zum Zitat Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain. J. Math. Fluid Mech. 15(1), 1–40 (2013)MathSciNetCrossRefMATH Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain. J. Math. Fluid Mech. 15(1), 1–40 (2013)MathSciNetCrossRefMATH
23.
Zurück zum Zitat Y. Shibata, S. Shimizu, On the maximal L p -L q regularity of the Stokes problem with first order boundary condition; model problems. J. Math. Soc. Japan 64(2), 561–626 (2012)MathSciNetCrossRefMATH Y. Shibata, S. Shimizu, On the maximal L p -L q regularity of the Stokes problem with first order boundary condition; model problems. J. Math. Soc. Japan 64(2), 561–626 (2012)MathSciNetCrossRefMATH
24.
Zurück zum Zitat H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Deutscher Verlag der Wissenschaften, Berlin, 1978)MATH H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Deutscher Verlag der Wissenschaften, Berlin, 1978)MATH
25.
Zurück zum Zitat L.R. Volevich, Solvability of boundary problems for general elliptic systems. Am. Math. Soc. Translat. II. Ser. 67, 182–225 (1968)MATH L.R. Volevich, Solvability of boundary problems for general elliptic systems. Am. Math. Soc. Translat. II. Ser. 67, 182–225 (1968)MATH
Metadaten
Titel
Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity
verfasst von
Robert Denk
Tim Seger
Copyright-Jahr
2016
Verlag
Springer Basel
DOI
https://doi.org/10.1007/978-3-0348-0939-9_9