Necessary and sufficient conditions for local solvability of nonsolvable partial differential equations
The principal result of this paper is as follows. Let P be a partial differential operator with analytic coefficients, and let p be the principal symbol of P. Let Char p be the characteristic variety of P. Assume there is a \(\lambda \in {\mathbb{R}}^ N\setminus \{0\}\) such that \(p(0,...,0,\lambda)\neq 0\). Let \(g\in {\mathcal E}'({\mathbb{R}}^ N)\) be such that \(WF_ A(g)\cap Char p\neq \emptyset\) in some neighborhood of \(0\in {\mathbb{R}}^ N\). Then there is a neighborhood U of \(0\in {\mathbb{R}}^ N\) and a distribution \(v\in {\mathcal D}'(U)\) such that \(Pv=g\) on U. Other results of this type are obtained. These generalize results of Greiner-Kohn-Stein for the Hans Lewy operator. The principal technique is to relate the problem to the Boutet de Monvel/Guillemin theory of Toeplitz operators. The facilitates passing from microlocal theory to local theory.
scientific article; zbMATH DE number 3906943