WIAS Preprint No. 2723, (2020)

On the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$


  • Chill, Ralph
  • Meinlschmidt, Hannes
    ORCID: 0000-0002-5874-8017
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35B65 35J15 47A12


  • Bidomain system, j-subgradient, gradient system, FitzHugh--Nagumo model




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 Lp in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. 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 characterization of elements of the form domains inducing mixed boundary conditions.

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