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Anal Appl applications assume Banach space blems boundary conditions boundary value problem bounded classical solution coefficients coercive compact consider control problems convergence convex corresponding defined denote dependence derivatives DiBenedetto differential equations dimensional distribution domain dual space elastic equilibrium equibounded estimate example existence Fasano finite constant finite element finite element method formulation Frechet-differentiable free boundary problems given heat equation Hence Hilbert space implies integral inverse Lemma lim inf linear functionals Lipschitz continuous lower semicontinuous mapping Math mathematical maximal monotone maximum principle method Moreover mushy regions nonlinear norm numerical obtain operator optimal control parabolic equations phase piecewise Primicerio Proof Proposition 4.2 prove R.E. Showalter regularity respect satisfies smooth Sobolev space solve Stefan problem Stefan-like problems subset sufficiently small temperature theory two-phase Stefan problem unique variational inequality weak solution weakly zero