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Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions
A. N. Polkovnikov,
A. A. Shlapunov Siberian Federal University, Krasnoyarsk, Russia
Abstract:
Let
$D$ be an open connected subset of the complex plane
$\mathbb C$ with sufficiently smooth boundary
$\partial D$. Perturbing the Cauchy problem for the Cauchy–Riemann system
$\bar\partial u=f$ in
$D$ with boundary data on a closed subset
$S\subset\partial D$, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter
$\varepsilon\in(0,1]$ in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on
$\partial D\setminus S$, each of them has a unique solution in some appropriate Hilbert space
$H^+(D)$ densely embedded in the Lebesgue space
$L^2(\partial D)$ and the Sobolev–Slobodetskiĭ space
$H^{1/2-\delta}(D)$ for every
$\delta>0$. The corresponding family of the solutions
$\{u_\varepsilon\}$ 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
$\{u_\varepsilon\}$ 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.
Keywords:
Cauchy–Riemann operator, Cauchy problem, Zaremba problem, small parameter, Laplace equation.
UDC:
517.35+
517.53
MSC: 35R30 Received: 25.10.2016
DOI:
10.17377/smzh.2017.58.414
Fulltext:
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