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Elliptic boundary-value problems in the sense of Lawruk on Sobolev and Hörmander spaces. (English) Zbl 1353.35154

Ukr. Math. J. 67, No. 5, 764-784 (2015) and Ukr. Mat. Zh. 67, No. 5, 672-691 (2015).
Summary: We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spaces \(H^{s,\varphi}\), which form the refined Sobolev scale. The order of differentiation for these spaces is given by a real number \(s\) and a positive function \(\varphi\) slowly varying at infinity in Karamata’s sense. We consider this problem for an arbitrary elliptic equation \(Au = f\) in a bounded Euclidean domain \(\Omega\) under the condition that \(u \in H^{s,\varphi} (\Omega)\), \(s < \operatorname{ord} A\), and \(f \in L_2(\Omega)\). We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:

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