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


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.

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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.

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Correspondence to Hannes Meinlschmidt.

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

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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

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

Mathematics Subject Classification

  • 35B65
  • 35J15
  • 47A12