Parabolic Obstacle-Type Problems: Regularity, Existence, and Deviation

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The proposed project is aimed at studying several parabolic free boundary problems of obstacle-type. The research program includes issues concerning well-posedness of parabolic problems involving equations with a hysteretic discontinuity in the source term, the study of regularity properties of the free boundary and the qualitative behavior of solutions for the above-mentioned problems, and the derivation of functionals that give realistic fully computable upper bounds of the difference between the exact solutions of the parabolic Signorini problems and any approximative solution regardless of the approximation method.In recent years the proposed research directions become very popular in the world. The regularity topics belong to the mainstream of investigations. Existence and nonexistence results are also not evident, especially for problems with non-variational nature. The estimates of deviation from the exact solution (upper bounds) lie on the crossroad of analytical and numerical studies of free boundary problems.All the suggested problems are new and original. Notice also that for obstacle-type problems the most part of the classical parabolic techniques is not applicable. The reason for this is an absence of any a priori information about the regularity of the free boundary. So, we have to combine ideas from the calculus of variations, some geometrical observations, rescaling and blow-up techniques, analysis of caloric functions and various monotonicity formulas.