On the numerical range of second-order elliptic operators with mixed boundary conditions in \(L^p\)

Abstract

We consider second-order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on \(L^p\) in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in Chill et al. (C R Acad Sci Paris 342:909–914, 2006). Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterisation of elements of the form domains inducing mixed boundary conditions.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. 1.

    H. Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory.

  2. 2.

    W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Handbook of Differential Equations (C. M. Dafermos, E. Feireisl eds.), Elsevier/North Holland, 2004, pp. 1–85.

  3. 3.

    M. Biegert, On traces of Sobolev functions on the boundary of extension domains, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4169–4176.

    MathSciNet  Article  Google Scholar 

  4. 4.

    K. Brewster, D. Mitrea, I. Mitrea, and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally\((\varepsilon ,\delta )\)-domains and applications to mixed boundary problems, J. Funct. Anal. 266 (2014), no. 7, 4314–4421.

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. Carbonaro and O. Dragičević, Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients, J. Eur. Math. Soc. (2019), no. to appear.

  6. 6.

    R. Chill, E. Fašangová, G. Metafune, and D. Pallara, The sector of analyticity of the Ornstein-Uhlenbeck semigroup in\(L^p\)spaces with respect to invariant measure, J. London Math. Soc. 71 (2005), 703–722.

  7. 7.

    R. Chill, E. Fašangová, G. Metafune, and D. Pallara, The sector of analyticity of nonsymmetric submarkovian semigroups generated by elliptic operators, C. R. Acad. Sci. Paris 342 (2006), 909–914.

    MathSciNet  Article  Google Scholar 

  8. 8.

    M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloquium Mathematicae 60-61 (1990), no. 2, 601–628 (eng).

  9. 9.

    A. Cialdea and V. Maz’ya, Criterion for the\(L^p\)-dissipativity of second order differential operators with complex coefficients, J. Math. Pures Appl. (9) 84 (2005), no. 8, 1067–1100.

  10. 10.

    R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.

    MathSciNet  Article  Google Scholar 

  11. 11.

    D. Daners, A priori estimates for solutions to elliptic equations on non-smooth domains, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), 793–813.

    MathSciNet  Article  Google Scholar 

  12. 12.

    E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  13. 13.

    R. Denk, M. Hieber, and J. Prüss, \({\cal{R}}\)-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Memoirs Amer. Math. Soc., vol. 166, Amer. Math. Soc., Providence, R.I., 2003.

  14. 14.

    M. Egert, R. Haller-Dintelmann, and J. Rehberg, Hardy’s inequality for functions vanishing on a part of the boundary, Potential Anal. 43 (2015), no. 1, 49–78.

    MathSciNet  Article  Google Scholar 

  15. 15.

    M. Egert and P. Tolksdorf, Characterizations of Sobolev functions that vanish on a part of the boundary, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 4, 729–743.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    J. A. Griepentrog, H.-C. Kaiser, and J. Rehberg, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on\(L^p\), Adv. Math. Sci. Appl. 11 (2001), no. 1, 87–112.

    MathSciNet  MATH  Google Scholar 

  17. 17.

    P. Hajłasz, P. Koskela, and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008), no. 5, 1217–1234.

    MathSciNet  Article  Google Scholar 

  18. 18.

    A. Jonsson and H. Wallin, Function spaces on subsets of\({{\bf R}}^n\), Math. Rep. 2 (1984), no. 1, xiv+221.

  19. 19.

    N. Kalton and L. Weis, The\(H^\infty \)-calculus and sums of closed operators, Math. Ann. 321 (2001), 319–345.

    MathSciNet  Article  Google Scholar 

  20. 20.

    P. C. Kunstmann, \(L_p\)-spectral properties of the Neumann Laplacian on horns, comets and stars, Math. Z. 242 (2002), 183–201.

    MathSciNet  Article  Google Scholar 

  21. 21.

    P. C. Kunstmann and L. Weis, Maximal\(L^p\)regularity for parabolic equations, Fourier multiplier theorems and\(H^\infty \)functional calculus, Levico Lectures, Proceedings of the Autumn School on Evolution Equations and Semigroups (M. Iannelli, R. Nagel, S. Piazzera eds.), vol. 69, Springer Verlag, Heidelberg, Berlin, 2004, pp. 65–320.

  22. 22.

    Damien Lamberton, Équations d’évolution linéaires associées à des semi-groupes de contractions dans les espaces\(L^p\), J. Funct. Anal. 72 (1987), no. 2, 252–262.

  23. 23.

    A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995.

    Google Scholar 

  24. 24.

    V. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, augmented ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011.

  25. 25.

    V. G. Maz’ya and S. V. Poborchiĭ, Theorems for embedding Sobolev spaces on domains with a peak and on Hölder domains, Algebra i Analiz 18 (2006), no. 4, 95–126.

    Google Scholar 

  26. 26.

    A. J. Morris, The Kato square root problem on submanifolds, J. London Math. Soc. 86 (2012), no. 3, 879–910.

    MathSciNet  Article  Google Scholar 

  27. 27.

    E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs, vol. 30, Princeton University Press, Princeton, 2004.

    Google Scholar 

  28. 28.

    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Berlin, 1983.

  29. 29.

    J. Prüss and G. Simonett, Moving interfaces and quasilinear parabolic evolution equations, Monographs in Mathematics, vol. 105, Birkhäuser/Springer, [Cham], 2016.

    Google Scholar 

  30. 30.

    A. F. M. ter Elst, R. Haller-Dintelmann, J. Rehberg, and P. Tolksdorf, On the\(L^p\)-theory for second-order elliptic operators in divergence form with complex coefficients, Preprint (2019), arXiv:1903.06692.

  31. 31.

    A. F. M. ter Elst, M. Meyries, and J. Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Ann. Mat. Pura Appl. (4) 193 (2014), no. 5, 1295–1318.

    MathSciNet  Article  Google Scholar 

  32. 32.

    H. Triebel, A note on function spaces in rough domains, Tr. Mat. Inst. Steklova 293 (2016), no. Funktsional. Prostranstva, Teor. Priblizh., Smezhnye Razdely Mat. Anal., 346–351, English version published in Proc. Steklov Inst. Math. 293 (2016), no. 1, 338–342.

  33. 33.

    A. Ukhlov, Extension operators on Sobolev spaces with decreasing integrability, Preprint (2019), arXiv:1908.09322.

Download references

Acknowledgements

We wish to thank Pertti Mattila (University of Helsinki) and Moritz Egert (Université Paris-Sud) for pointing out the Christ decomposition and ideas for the proof of Lemma 6.6. We are also grateful to the anonymous referee for their useful comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hannes Meinlschmidt.

Additional information

Dedicated to Matthias Hieber on the occasion of his 60th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chill, R., Meinlschmidt, H. & Rehberg, J. On the numerical range of second-order elliptic operators with mixed boundary conditions in \(L^p\). J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00642-6

Download citation

Keywords

  • Elliptic operator
  • Resolvent estimates
  • Numerical range
  • Mixed boundary conditions
  • Nonsmooth domains
  • Ultracontractivity
  • Dynamic boundary conditions
  • Intrinsic characterisation

Mathematics Subject Classification

  • 35B65
  • 35J15
  • 47A12