Linear Partial Differential Equations of Second Order

Abstract

Let us consider the differential operator

$$L(u)(x)\!=\!\sum _{i, j=1}^{n}a_{ij}(x)\frac{\partial ^{2} u}{\partial x_{i}\partial x_{j}}(x)\!+\! \sum _{j=1}^{n}a_{j}(x)\frac{\partial u}{\partial x_{j}}(x)\!+\!a_{0}(x)u(x),$$

in which the coefficients $a_{ij}\;(i, j=\overline{1,n}),\; a_{j}\,(j=\overline{1,n})$ and $a_{0}$ are regular functions which depend only on one variable x and which are defined on an open set $\Omega$ from $R^{n}$, where $\Omega$ is not necessarily bounded.