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Regularity estimates for the oblique derivative problem. (English) Zbl 0792.35039

The authors consider the oblique derivative problem: \(Pu = f\) in \(\Omega\), 1\(u = 0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(P\) is a second order elliptic differential operator and 1 is a real unit vector field. They study the \(L_ p\)-Sobolev regularity of the solution \(u\) assuming that the field 1 may be tangent to \(\partial \Omega\) on a part of it, but is such that the solution is regular if \(f\) is regular [J. V. Egorov and V. A. Kondrat’ev, Mat. Sb., Nov. Ser. 78(120), 148-176 (1969; Zbl 0165.122)]. They prove that \(u\) does not gain 2 derivatives from \(f\) in \(L_ p\)-Sobolev spaces, but does gain \(2-\varepsilon\) derivatives for all \(\varepsilon > 0\). Moreover in the case when 1 never crosses \(\partial \Omega\), then one has the gain of 2 derivatives iff 1 is of finite type, and in the case when 1 crosses \(\partial \Omega\) along a submanifold \(\Gamma\), then the necessary and sufficient condition involves a one-sided version of the Muckenhoupt condition \(A_ p\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0165.122
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