On fundamental solutions of a class of weak hyperbolic operators
Views: 16 / PDF downloads: 5
Keywords:
hyperbolic with weight operator (polynomial), multianisotropic Jevre space, Newton polyhedron, fundamental solutionAbstract
We consider a certain class of polyhedrons ℜ⊆En, multi-anisotropic Jevre spaces Gℜ(En), their subspaces Gℜ0(En), consisting of all functions f∈Gℜ(En) with compact support, and their duals (Gℜ0(En))∗. We introduce the notion of a linear differential operator P(D), hℜ-hyperbolic with respect to a vector N∈En, where hℜ is a weight function generated by the polyhedron ℜ. The existence is shown of a fundamental solution E of the operator P(D) belonging to (Gℜ0(En))∗ with suppE⊆ΩN, where ΩN:={x∈En,(x,N)>0}. It is also shown that for any right-hand side f∈Gℜ(En) with the support in a cone contained in ΩN and with the vertex at the origin of En, the equation P(D)u=f has a solution belonging to Gℜ(En).