Pseudodifferential equations in unbounded regions with conical structure at infinity

V. S. Rabinovich

Sbornik. Mathematics, Tome 9 (1969) no. 1, pp. 73-92

Résumé

We consider pseudodifferential equations of the form \begin{equation} Au\equiv\int_G a(x,x-y)u(y)\,dy=f(x),\qquad x\in G, \tag{1} \end{equation} where $G$ is an unbounded region in $R^n$ which has a smooth boundary $\partial G$ and which is a conical set outside a sphere of sufficiently large radius. The symbol $\widetilde a(x,\xi)$ of the pseudodifferential operator $A$ is either a function which is continuous with respect to $\xi$ on $R^n_\xi$, which is the extension of $R^n_\xi$ obtained by adding a point at infinity, or is a function having polynomial growth as $|\xi|\to\infty$. With respect to $x$ the symbol is bounded, satisfies certain smoothness conditions, and is not necessarily stabilized as $x\to\infty$. We study (1) in the Sobolev–Slobodetskii functional spaces $H^s$. Depending on $s$, the equation (1) becomes a properly posed problem either with general boundary conditions or with additional potentials. For some $s$ we can regard (1) as an integral equation which needs no additional conditions. We will obtain necessary and sufficient conditions under which properly posed problems for the equation (1) will be Noetherian in the Sobolev–Slobodetskii spaces. Bibliography: 17 titles.