Colloquium April 2
Date: Thursday, April 2
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Luigi Berselli of Università di Pisa
Title: On the existence of weak, strong and very-weak solutions. From the Poisson problem to the Stokes equations
Abstract: We consider the variational formulation of boundary value problems for the Stokes equations and we study the existence and uniqueness of weak, strong, and very-weak solutions. The first two classes of solutions are very well-known, while the last one is the object of ongoing research. The concept of very-weak solution is of particular interest in applications to shape optimization, problems with rough boundaries, and also in problems of numerical error estimations.
To introduce the problems related to the very-weak solution we review the results on the same class of solutions for the Poisson problem, i.e., the partial differential equations related to the Laplace operator with various boundary conditions and we review with the basic existence and regularity theorems. Finally, we consider the Stokes problem with Dirichlet and Navier boundary conditions. Especially the last case is of interest because many classes of solutions can be treated with elementary tools since we will show how to reduce this problem to the solution of suitable Poisson problems.
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