Math 6410 • ODE

Elliptic Equations. Suppose \(\Omega \subset \mathbb {R}^n\) is a smooth bounded domain. Let \(a^i(x)\in C^\infty (\overline {\Omega })\) be smooth functions. Give conditions on \(a^i(x)\) so that for all \(f\in L^2(\Omega )\) there exists a weak solution to the boundary value problem \[ \begin {cases} \Delta u+\displaystyle \sum _{i=1}^n a^i(x)D_i u=f(x), & \text {for }x\in \Omega ,\\ u(x)=0, & \text {for }x\in \partial \Omega . \end {cases} \] [Here \(\Delta =\sum _{i=1}^n\frac {\partial ^2}{\partial x_i^2}\) is the Laplacian and \(D_i=\frac {\partial }{\partial x_i}\) is the partial derivative.]