Istituto di Matematica Applicata e Tecnologie Informatiche     
Gianazza U., Stroffolini B., Vespri V. Interior and Boundary Continuity of the Solution of the Singular Equation $(beta(u))_t=Lu$. Preprint ercim.cnr.ian//2002-1310, 2002.
We deal with the singular parabolic equation $$(beta (u))_{t}={cal L} u,$$ of the kind arising in the modelling of phase transition for fluids and prove interior and boundary continuity of the weak solutions. With respect to previous results due to DiBenedetto and Vespri where ${cal L}$ is simply the Laplace operator, here we consider a more general situation, where ${cal L} $ is a second order elliptic operator with bounded and measurable coefficients that depend both on space and time in a proper way. The main focus is on the interior behaviour, where special care has to be used in order to deal with the lack of radial simmetry of ${cal L}$. We first develop the case of time - independent coefficients and then consider the general case by a perturbation argument. With regards to the boundary behaviour, we deal with homogeneous Dirichlet conditions. This is simply a first step towards a comprehensive study of general Dirichlet and Neumann conditions for this kind of equations.
Subject Singular parabolic equation, continuity, boundary conditions
35R35, 35K55

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