On the absolute convergence of spectral expansions in a two-dimensional closed region for the Laplace operator with a discontinuous coefficient and the Dirichlet problem

The article examines the Laplace operator with a discontinuous coefficient (piecewise constant) and the Dirichlet problem for a closed region. The discontinuity of the coefficient occurs on some surface located inside the region. As was shown earlier, if the dimensionality of the region $N$ is large enough, for example, $N\geqslant 5$, then the presence of such a discontinuity does not guarantee the convergence of spectral decompositions even in regions "far" from the coefficient discontinuity points for functions arbitrarily smooth and finite with respect to the considered region. This work shows that if the dimensionality of the region $N$ is equal to two, then the presence of such a discontinuity does not affect the absolute convergence of the spectral expansions in the whole closed region containing the discontinuity points.