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Suppose that the set of square-integrable solutions of an elliptic equation which have a value at some given point equal to c is not empty. Then there is exactly one element with minimal L 2 {L^{2}} -norm. Moreover, it is shown that this minimal element depends continuously on a domain of integration, i.e., on the set on which our solutions are defined, and on a weight of integration, i.e., on the deformation of an inner product. The theorems are proved using the theory of reproducing kernels and Hilbert spaces of square-integrable solutions of elliptic equations. We prove the existence of such a reproducing kernel using theory of Sobolev spaces. We generalize the well-known Ramadanov theorem. This is done in three different ways. Two of them are similar to the techniques used by I. Ramadanov and M. Skwarczyński(see [11, 14, 13]) , while the third method using weak convergence is new. Moreover, we show that our reproducing kernel depends continuously on a weight of integration. The idea of using the minimal norm property in such a proof is novel and, which is important, it needs the convergence of weights only almost everywhere.

In this paper we consider spaces of weight square-integrable and harmonic functions L 2 H(Ω, µ). Weights µ for which there exists reproducing kernel of L 2 H(Ω, µ) are named ’admissible weights’ and such kernels are named ’harmonic Bergman kernels’. We prove that if only weight of integration is integrable in some negative power, then it is admissible. Next we construct a weight µ on the unit circle which is non-admissible and using Bell-Ligocka theorem we show that such weights exist for a large class of domains in ℝ2. Later we conclude from the classical result of reproducing kernel Hilbert spaces theory that if the set {f ∈ L 2 H(Ω, µ)|f(z) = c} for admissible weight µ is non-empty, then there is exactly one element with minimal norm. Such an element in this paper is called ’a minimal (z, c)-solution in weight µ of Laplace’s equation on Ω’ and upper estimates for it are given.

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