On the numerical range of second-order elliptic operators with mixed boundary conditions in $$L^p$$ L p
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Journal of Evolution Equations
On the numerical range of second-order elliptic operators with mixed boundary conditions in L p Ralph Chill, Hannes Meinlschmidt
and Joachim Rehberg
Dedicated to Matthias Hieber on the occasion of his 60th birthday. 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.
1. Introduction The regularity of solutions of elliptic or parabolic operators is a classical subject. Uniform estimates for resolvents of elliptic operators and for the semigroups generated by them are central instruments for the study of nonautonomous linear or quasilinear parabolic equations. Much of the theory is standard nowadays and treated in many comprehensive books on parabolic evolution equations; we refer exemplarily to [1, Chapter II], [23, Chapter 6.1], [13] or [29]. In this work, we provide uniform resolvent estimates for the L p -realizations of second-order elliptic operators with real, nonsymmetric coefficient functions posed on bounded domains in Rd and subject to mixed boundary conditions, under minimal regularity assumptions on the domain. In case of smooth domains and real and symmetric coefficient functions, such uniform resolvent estimates are classical ( [28, Chapter 7.3]) and have been generalized in [16] to nonsmooth domains and mixed boundary conditions. Moreover, the case of nonsymmetric coefficient functions has been treated in [7] under pure Dirichlet or pure Neumann or Robin conditions. Mathematics Subject Classification: 35B65, 35J15, 47A12 Keywords: Elliptic operator, Resolvent estimates, Numerical range, Mixed boundary conditions, Nonsmooth domains, Ultracontractivity, Dynamic boundary conditions, Intrinsic characterisation.
R. Chill et al.
J. Evol. Equ.
Our main result is a complement to the main result in [7]. We give an (optimal) estimate for the half angle θ p of the sector containing the numerical range of the L p -realization of the elliptic operator. The proof given here differs from the proof in [7] and uses ideas from [9]. The estimate for the numerical range immediately yields resolvent estimates outside the sector with half angle θ p , and these estimates stand in a one-to-one correspondence to the holomorphy of the corresponding semigroup on a sector with half angle π2 − θ
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