Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

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 L ₂(∂D) and the Sobolev–Slobodetskiĭ space H ^1/2−δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} 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 _ε} 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.