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Interpolation for function spaces related to mixed boundary value problems. (English) Zbl 1010.46021
For $$1<p< \infty$$, $$|s|\leq 1$$ and $$\Omega$$ a bounded Lipschitz domain in $${\mathbb{R}}^d$$, let $$H^{s,p }(\Omega)$$ denote the corresponding Sobolev space and, if $$\Gamma$$ is a part of the boundary $$\partial \Omega$$, let $$H^{s,p }_{\Gamma}(\Omega)$$ be the subspace of all functions with vanishing trace on $$\Omega \cap \Gamma$$. Under some natural restrictions on the parameters and on $$\Gamma$$, for the complex interpolation method the authors show that $[H^{s_0,p }_{\Gamma}(\Omega),H^{s_1,p }_{\Gamma}(\Omega)]_\theta =H^{s,p }_{\Gamma}(\Omega) \qquad(s=(1-\theta)s_0+\theta s_1))$ by using some adequate retraction and contraction operators to obtain a reduction to the well known case of couples $$(H_0^{s_0,p }(B),H_0^{s_1,p }(B))$$ on the unit ball $$B$$.

##### MSC:
 46B70 Interpolation between normed linear spaces 46M35 Abstract interpolation of topological vector spaces 35J25 Boundary value problems for second-order elliptic equations
##### Keywords:
Sobolev spaces; complex interpolation
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