Reproducing kernels and minimal solutions of elliptic equations

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-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. Skwarczynski, 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.