On the general theory of boundary value problems

In a bounded domain $V$ in $n$-dimensional Euclidean space each formal, linear, partial differential operator $L(D)$ with constant coefficients may be connected with so-called minimal $L_0$ and maximal $\widetilde L$ operators in the Hilbert space $\mathscr L^2(V)$. The operator $L$ is said to be proper if $L_0\subset L\subset\widetilde L$ and the equation $Lu=f$ has a unique solution for any $f\in\mathscr L^2(V)$. Using the complete description of proper operators that we obtain for $n=1$, in this article we discuss problems connected with the description of proper operators in the general case when $n>1$.
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