Dirichlet and Neumann boundary conditions: What is in between?

Given an admissible measure mu on partial derivativeOmega where Omega subset of R(n) is an open set, we define a realization Delta(mu) of the Laplacian in L(2)(Omega) with general Robin boundary conditions and we show that Delta(mu) generates a holomorphic C(0)-semigroup on L(2)(Omega) which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form Delta(mu). We also show that if D(Delta(mu)) contains smooth functions, then mu is of the form dmu = betadsigma (where sigma is the (n - 1)-dimensional Hausdorff measure and beta a positive measurable bounded function on a partial derivativeOmega); i.e. we have the classical Robin boundary conditions.