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Can the non divergence form operator $\mathcal{L}u= u_{xx}+u_{yy} + u_x=\Delta u + u_x$ be put in divergence form? In general, can any constant coefficient non divergence form operator be put into divergence form?

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Yes. To put an operator into divergence form we multiply $Lu$ by a function $\varphi $ and integrate by parts to get rid of the second derivatives of $u$. This works as long as the coefficients of second derivatives are sufficiently smooth, and constant functions are very smooth.

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