Conditions for the solvability of boundary value problems for quasi-elliptic systems in a half-space
The author considers boundary value problems in \(\mathbb R^n_+=\{x=(x',x_n)\;:\;x'\in\mathbb R^{n-1},\;x_n>0\}\) for quasi-elliptic systems of the form \[ \mathcal L(D_x)U=F(x),\quad x\in\mathbb R^n_+,\qquad \mathcal B(D_x)U|_{x_n=0}=0.\tag{1} \] Conditions, imposed on the matrix differential operator \(\mathcal L(D_x)\), define it as a quasi-elliptic operator introduced by \textit{L. R. Volevich} [Mat. Sb., N. Ser. 59(101), Suppl., 3--52 (1962; Zbl 0161.07801)], and conditions, imposed on the matrix differential operator \(\mathcal B(D_x)\), make the problem (1) satisfying the Lopatinskii condition. The author obtains necessary and sufficient conditions for unique solvability of the problem in appropriate Sobolev spaces.
scientific article; zbMATH DE number 6065260