Variational Formulation

Abstract

Techniques based on classical function spaces are less suited for proving the existence of a solution of a boundary-value problem. Section 7.1 introduces another approach via a variational problem (Dirichlet’s principle). Combining the variational formulation with the Sobolev spaces will be successful. In Section 7.2 the boundary-value problem of order 2m with homogeneous Dirichlet conditions is transferred into the variational formulation in the space $H^{m}_{0}(\it\Omega)$. Existence of a solution in $H^{m}_{0}(\it\Omega)$ follows in Theorem 7.8 from the $H^{m}_{0}(\it\Omega)$-ellipticity which is discussed, e.g., in the Theorems 7.3 and 7.7. In Section 7.3 we consider inhomogeneous Dirichlet boundary-value problems. The natural boundary condition in Section 7.4 follows from variation in $H^{m}_{0}(\it\Omega)$ without any restrictions. In the case of the Poisson equation one obtains the Neumann condition, in the general case the conormal boundary derivative appears. We investigate how general boundary conditions can be formulated as variational problem. Complications appearing for differential equations of higher order are explained by taking the example of the biharmonic equation.