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

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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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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Keywords

  • mixed problems
  • noncoercive boundary conditions
  • elliptic operators
  • root functions
  • weighted Sobolev spaces