On fundamental solutions of a class of weak hyperbolic operators

V. N. Margaryan ; H. G. Ghazaryan

Eurasian mathematical journal, Tome 9 (2018) no. 2, pp. 54-67

Résumé

We consider a certain class of polyhedrons $\mathfrak{R}\subset\mathbb{E}^n$, multi-anisotropic Jevre spaces $G^{\mathfrak{R}}(\mathbb{E}^n)$, their subspaces $G_0^{\mathfrak{R}}(\mathbb{E}^n)$, consisting of all functions $f\in G^{\mathfrak{R}}(\mathbb{E}^n)$ with compact support, and their duals $(G_0^{\mathfrak{R}}(\mathbb{E}^n))^*$. We introduce the notion of a linear differential operator $P(D)$, $h_{\mathfrak{R}}$-hyperbolic with respect to a vector $N\in\mathbb{E}^n$, where $h_{\mathfrak{R}}$ is a weight function generated by the polyhedron $\mathfrak{R}$. The existence is shown of a fundamental solution $E$ of the operator $P(D)$ belonging to $(G_0^{\mathfrak{R}}(\mathbb{E}^n))^*$ with $\mathrm{supp}\, E\subset\overline{\Omega_N}$, where $\Omega_N:=\{x\in\mathbb{E}^n, (x, N)>0\}$. It is also shown that for any right-hand side $f\in G^{\mathfrak{R}}(\mathbb{E}^n)$ with the support in a cone contained in $\overline{\Omega_N}$ and with the vertex at the origin of $\mathbb{E}^n$, the equation $P(D)u = f$ has a solution belonging to $G^{\mathfrak{R}}(\mathbb{E}^n)$.