Unified theory for abstract parabolic boundary problems-a semigroup approach


This paper presents and abstract semigroup formulation of parabolic boundary value problems. Smoothness of solutions, represented by a semigroup formula, is the primary object of discussion. The generality of our approach enables us to treat in a unified manner the regularity of solutions to parabolic equations for a large variety of nonhomogeneous boundary value problems. In particular, the approach presented here allows us to translate known regularity results of the elliptic theory directly into regularity results for the parabolic solutions. On the one hand, our theory recaptures known regularity results of the parabolic solutions over smooth spatial domains. On the other hand, however, our theory also covers the case of conical spatial domains, for which the standard assumption of C∞-boundaries is violated by suitable application of recent relevant results of elliptic theory for such domains. In the concluding section, an application of our general theory to a boundary control problem with a quadratic performance index is presented. © 1980 Springer-Verlag New York Inc.

Publication Title

Applied Mathematics & Optimization