Abstract Using a relation between representation theory of crystallographic space groups and a Dirichlet type of boundary problem for the Laplacian, we derive the solutions for the Dirichlet problem, as well as for a similar Neumann boundary problem, by a complete decomposition of plane waves into irreducible representations of a particular space group. This decomposition corresponds to a basis transformation in L2(Ω) and yields a new set of basis functions adapted to the symmetry of the lattice considered.
We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. We construct an approximate solution to the problem via an elliptic approximation for which, under appropriate assumptions, we prove that the corresponding limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. We also provide numerical examples.
ON THE DIRICHLET-NEUMANN BOUNDARY PROBLEM FOR SCALAR CONSERVATION LAWS
