Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions

DOI10.1007/S00028-018-0467-XarXiv1707.07718WikidataQ129271416 ScholiaQ129271416MaRDI QIDQ6289376

Publication date: 24 July 2017

Abstract: Let

O m e g a

be a bounded open subset with

C^{1 + k a p p a}

-boundary for some

k a p p a > 0

. Consider the Dirichlet-to-Neumann operator associated to the elliptic operator

- s u m p a r t i a l_{l} (c_{k l}, p a r t i a l_{k}) + V

, where the

c_{k l} = c_{l k}

are H"older continuous and

V i n L_{i} n f t y (O m e g a)

are real valued. We prove that the Dirichlet-to-Neumann operator generates a

C_{0}

-semigroup on the space

C (p a r t i a l O m e g a)

which is in addition holomorphic with angle

f r a c p i 2

. We also show that the kernel of the semigroup has Poisson bounds on the complex right half-plane. As a consequence we obtain an optimal holomorphic functional calculus and maximal regularity on

L_{p} (G a m m a)

for all

p i n (1, i n f t y)

.

Mathematics Subject Classification ID

One-parameter semigroups and linear evolution equations (47D06) Heat kernel (35K08)

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