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If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ is a sufficiently smooth diffeomorphism to another domain $W$, does the PDE that the transformed solution $\tilde u = u \circ T^{-1}$ (which lives on $W$) satisfies, also satisfy the Lopatinski-Shapiro condition?

Explicitly calculating the condition for the transformed version is very very messy, so this result would be handy.

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Yes, according to Agranovich, Elliptic boundary value problems, Encyclopedia of Mathematical Sciences, Springer 1997. See top of page 8.

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