Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators
Let \(\Omega\) be a domain in \(\mathbb R^n\). Under consideration is the problem \[ L(x,D_x)u(x)=f(x)\;(x\in \Omega),\;\;B(x,D_x)u(x)=g(x)\;(x\in \partial \Omega), \] with \(L\) a matrix operator whose entries are differential operators and \(B(x,D_x)\) some collection of operators. It is assumed that \(L\) is symmetric and the expression \((Lu,v)\) can be written as \(a(u,v)\) with \(a\) a symmetric sesquilinear form. The author studies a special polynomial property for the operator \(L\) which actually means that the form \(a\) vanishes only on a finite-dimensional subspace of vector polynomials. It is shown that the problem is an elliptic problem whenever, for every \(x_0\in \overline{\Omega}\), the operator \(L(x_0,D_x)\) with frozen coefficients possesses the polynomial property. Next, in terms of the above subspace of polynomials, the author describes a kernel, co-kernel of the problem, and solvability conditions for this problem.
unbounded domains with special exits at infinity
scientific article; zbMATH DE number 1495302
scientific article