Abstract We consider optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems and obtain the general criterion for a very weak solution to be regular in the neighborhood of a given point. First, by Hodge decomposition and the technique of filling holes, we establish the relation between the very weak solution and the classical weak solution. Furthermore, combining the technique of p -harmonic approximation with the method of Hodge decomposition, we obtain the partial regularity result. In particular, the partial regularity we obtained is optimal. 

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Optimal partial regularity for very weak solutions to a class of nonlinear elliptic systems

Null Controllability of Networks Systems

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.

G. Leugering

Papers

Optimal control problems of parabolic fractional sturm liouville equations in a star graph