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

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  • Why study these equations in divergent form?
  • Why the minus sign in the principal part?

Thanks very much

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  • $\begingroup$ The short answer is: "keep on reading, you'll find out yourself". Anyway, the minus is just a convention, while the divergence form is better suited to the integration by parts that is surely coming a couple of lines below that excerpt. $\endgroup$ – Giuseppe Negro May 12 '16 at 14:30
  • $\begingroup$ 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? $\endgroup$ – C. Junior May 12 '16 at 14:37
  • $\begingroup$ 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. $\endgroup$ – Ian May 12 '16 at 14:46
  • $\begingroup$ Also use the search function: math.stackexchange.com/questions/1772588/… $\endgroup$ – Andrew May 12 '16 at 18:25

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