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

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.