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Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

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Abstract

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rn for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ⊂ ∂D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, 660041, Russia

    A. A. Shlapunov

  2. Institut für Mathematik, Universität Potsdam, Potsdam, 14469, Germany

    N. N. Tarkhanov

Authors
  1. A. A. Shlapunov
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  2. N. N. Tarkhanov
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Correspondence to A. A. Shlapunov.

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Original Russian Text ©A. A. Shlapunov and N. Tarkhanov, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 2, pp. 133–204.

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Shlapunov, A.A., Tarkhanov, N.N. Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II. Sib. Adv. Math. 26, 247–293 (2016). https://doi.org/10.3103/S1055134416040027

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