# Weak solutions for elliptic equations

i have two questions about the study of weak solutions for elliptic equations?

• Why study these equations in divergent form?
• Indeed, the divergence form helps the integration by parts. If $a_{ij}$ is $C^{1}$, then the divergence and non-divergence form is equivalent. What is the convention you speak? – C. Junior May 12 '16 at 14:37
• The minus sign makes it so that if $A$ is a positive definite matrix then the bilinear form corresponding to $u \mapsto \nabla \cdot (a \nabla u)$ is positive definite. Without the minus sign it is negative definite, which is fine but creates more minus signs further down the line. – Ian May 12 '16 at 14:46