Advertisement

Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators

  • A. F. M. ter ElstEmail author
  • M. F. Wong
Article
  • 10 Downloads

Abstract

Consider the elliptic operator
$$\begin{aligned} A = - \sum _{k,l=1}^d \partial _k \, c_{kl} \, \partial _l + \sum _{k=1}^d a_k \, \partial _k - \sum _{k=1}^d \partial _k \, b_k + a_0 \end{aligned}$$
on a bounded connected open set \(\Omega \subset \mathbb {R}^d\) with Lipschitz boundary conditions, where \(c_{kl} \in L_\infty (\Omega ,\mathbb {R})\) and \(a_k,b_k,a_0 \in L_\infty (\Omega ,\mathbb {C})\), subject to Robin boundary conditions \(\partial _\nu u + \beta \, {\mathop {\mathrm{Tr \,}}}u = 0\), where \(\beta \in L_\infty (\partial \Omega , \mathbb {C})\) is complex valued. Then we show that the kernel of the semigroup generated by \(-A\) satisfies Gaussian estimates and Hölder Gaussian estimates. If all coefficients and the function \(\beta \) are real valued, then we prove Gaussian lower bounds. Finally, if \(\Omega \) is of class \(C^{1+\kappa }\) with \(\kappa > 0\), \(c_{kl} = c_{lk}\) is Hölder continuous, \(a_k = b_k = 0\) and \(a_0\) is real valued, then we show that the kernel of the semigroup associated to the Dirichlet-to-Neumann operator corresponding to A has Hölder Poisson bounds.

Keywords

Robin boundary conditions Dirichlet-to-Neumann operator Heat kernel estimates 

Mathematics Subject Classification

35K05 35B45 35J25 

Notes

Acknowledgements

The authors wish to thank Wolfgang Arendt for many discussions at various stages of this project. The authors wish to thank Mourad Choulli for helpful comments regarding the chain condition. Moreover, the authors wish to thank the referees for their comments. The first-named author is most grateful for the hospitality extended to him during a fruitful stay at the University of Ulm. He wishes to thank the University of Ulm for financial support. Part of this work is supported by an NZ-EU IRSES counterpart fund and the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Part of this work is supported by the EU Marie Curie IRSES Program, Project ‘AOS’, No. 318910.

References

  1. 1.
    Arendt, W. and Elst, A.F.M. ter, Gaussian estimates for second order elliptic operators with boundary conditions. J. Oper. Theory 38 (1997), 87–130.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arendt, W. and Elst, A.F.M. ter, Sectorial forms and degenerate differential operators. J. Oper. Theory 67 (2012), 33–72.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arendt, W. and Elst, A.F.M. ter, The Dirichlet problem without the maximum principle. Annales de l’Institut Fourier 69 (2019), 763–782.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arendt, W., Elst, A.F.M. ter, Kennedy, J. B. and Sauter, M., The Dirichlet-to-Neumann operator via hidden compactness. J. Funct. Anal. 266 (2014), 1757–1786.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aronson, D. G., Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73 (1967), 890–896.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Auscher, P., Regularity theorems and heat kernels for elliptic operators. J. Lond. Math. Soc. 54 (1996), 284–296.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Auscher, P. and Tchamitchian, P., Gaussian estimates for second order elliptic divergence operators on Lipschitz and $C^1$ domains. In Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), vol. 215 of Lecture Notes in Pure and Appl. Math., 15–32. Marcel Dekker, New York, 2001.CrossRefGoogle Scholar
  8. 8.
    Behrndt, J. and Elst, A. F. M. ter, Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps. J. Spectr. Theory  (2019). In press.Google Scholar
  9. 9.
    Choi, J. and Kim, S., Green’s functions for elliptic and parabolic systems with Robin-type boundary conditions. J. Funct. Anal. 267 (2014), 3205–3261.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Choulli, M. and Kayser, L., Gaussian lower bound for the Neumann Green function of a general parabolic operator. Positivity 19 (2015), 625–646.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Daners, D., Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217 (2000), 13–41.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Daners, D., Inverse positivity for general Robin problems on Lipschitz domains. Arch. Math. 92 (2009), 57–69.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Davies, E. B., Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92. Cambridge University Press, Cambridge etc., 1989.Google Scholar
  14. 14.
    Elst, A.F.M. ter and Ouhabaz, E.-M., Partial Gaussian bounds for degenerate differential operators II. Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (2015), 37–81.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Elst, A.F.M. ter and Ouhabaz, E.-M., Poisson bounds for the Dirichlet-to-Neumann operator on a $C^{1+\kappa }$-domain. J. Differ. Equ. 267 (2019), 4224–4273.CrossRefGoogle Scholar
  16. 16.
    Elst, A.F.M. ter and Rehberg, J., Hölder estimates for second-order operators on domains with rough boundary. Adv. Differ. Equ. 20 (2015), 299–360.zbMATHGoogle Scholar
  17. 17.
    Elst, A.F.M. ter and Robinson, D.W., Local lower bounds on heat kernels. Positivity 2 (1998), 123–151.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions. Studies in advanced mathematics. CRC Press, Boca Raton, 1992.zbMATHGoogle Scholar
  19. 19.
    Gesztesy, F., Mitrea, M. and Nichols, R., Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions. J. Anal. Math. 122 (2014), 229–287.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies 105. Princeton University Press, Princeton, 1983.Google Scholar
  21. 21.
    Kato, T., Perturbation theory for linear operators. Second edition, Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag, Berlin etc., 1980.Google Scholar
  22. 22.
    Nittka, R., Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equ. 251 (2011), 860–880.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ouhabaz, E.-M., Analysis of heat equations on domains, vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

Personalised recommendations