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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 \(B\).

46B70 Interpolation between normed linear spaces
46M35 Abstract interpolation of topological vector spaces
35J25 Boundary value problems for second-order elliptic equations
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