Spectral asymptotic and positivity for singular Dirichlet-to-Neumann operators

In the framework of Hilbert spaces we shall give necessary and sufficient conditions to define a Dirichlet-to-Neumann operator via Dirichlet principle. Analyzing singular operators, will establish Laurent expansion near singularities as well Mittag–Leffler for related quadratic forms. The established results be exploited solve definitively problem positivity semigroup in Lebesgue spaces. obtained are supported by some examples where analyze properties operators Neumann Robin Laplacian on Lipschitz domains. Among other results, demonstrate that regularity boundary may affect derive Mittag-Leffler eigenvalues operators.