Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions
- Authors: Polkovnikov A.N.1, Shlapunov A.A.1
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Affiliations:
- Siberian Federal University
- Issue: Vol 58, No 4 (2017)
- Pages: 676-686
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/171350
- DOI: https://doi.org/10.1134/S0037446617040140
- ID: 171350
Cite item
Abstract
Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ∂D. Perturbing the Cauchy problem for the Cauchy–Riemann system ∂̄u = f in D with boundary data on a closed subset S ⊂ ∂D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ∂D\S, each of them has a unique solution in some appropriate Hilbert space H+(D) densely embedded in the Lebesgue space L2(∂D) and the Sobolev–Slobodetskiĭ space H1/2−δ(D) for every δ > 0. The corresponding family of the solutions {uε} converges to a solution to the Cauchy problem in H+(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H+(D) is equivalent to boundedness of the family {uε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.
About the authors
A. N. Polkovnikov
Siberian Federal University
Author for correspondence.
Email: paskaattt@yandex.ru
Russian Federation, Krasnoyarsk
A. A. Shlapunov
Siberian Federal University
Email: paskaattt@yandex.ru
Russian Federation, Krasnoyarsk