Variational aspects of phase transitions with prescribed mean curvature

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These are solutions to an inhomogeneous semilinear elliptic PDE that give rise diffuse objects (varifolds) limit hypersurfaces, possibly singularities, whose is determined by "prescribed curvature" function and limiting multiplicity. establish upper bounds for eigenvalues problem, as well more subtle lower when problem converges multiplicity one. For latter, we also asymptotics sharp order $o(\varepsilon^2)$ $C^{2,\alpha}$ estimates on multiplicity-one transition layers.