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For an order-$m$ operator with smooth coefficients on a bounded piecewise smooth domain, integration by parts has no boundary remainder when at each boundary point either the entire boundary jet of $u$ through order $m-1$ vanishes or the entire such jet of $v$ vanishes. Every boundary summand pairs derivatives of the two functions of order at most $m-1$. It is insufficient to require only that, for each same multi-index $\beta$, one of ${\partial }^{\beta }u$ and ${\partial }^{\beta }v$ vanish: for $P={\partial }_x^2$, $u=x(a-x)\chi (y)$ and $v=1$ on a rectangle, all those same-index products vanish on the boundary but $\int (Pu)v=-2a\int \chi \ne 0$.

Near a non-characteristic plane $x=0$, use the small cap ${\omega }_{\epsilon }=\{x>0,x+y^2<\epsilon \}$. Its curved face is non-characteristic by continuity of the principal symbol. A polynomial coordinate change flattens that face. Uniform real analytic coefficient bounds and the uniform Cauchy radius for polynomial forcing solve $P^{\ast }v=g$ throughout a common cap, with zero boundary jet on the curved face. The complete boundary-jet condition for formal adjoints then gives ${\int }_{{\omega }_{\epsilon }}ug=0$ for every polynomial $g$. Uniform approximation of $\bar{u}$ gives $\int |u{|}^2=0$. Repeating on the other side of the plane proves two-sided local uniqueness.

In the flattened coordinates, ${\partial }_x={\partial }_{\tilde{x}}$ and ${\partial }_y=2\tilde{y}{\partial }_{\tilde{x}}+{\partial }_{\tilde{y}}$. The transformed formal adjoint consequently has coefficients obtained from finitely many real analytic coefficient pullbacks and polynomial factors. They have uniform analyticity bounds on a fixed small ball. By the preceding non-characteristic argument, its coefficient of ${\partial }_{\tilde{x}}^m$ has modulus bounded away from zero there, uniformly for small $\epsilon$. Division by it preserves uniform real analytic bounds.

The transformed forcing ${\tilde{g}}_{\epsilon }$ has analogous bounds, with constants that may depend on $g$. Apply the Cauchy-Kovalevskaya theorem in normal form with zero normal data through order $m-1$ at $\tilde{x}=0$. The supplied majorant series radius result yields a common solution radius $r_g>0$ for all sufficiently small parameters. Tangential differentiation of zero normal data shows that the entire boundary jet through order $m-1$ is zero; changing coordinates preserves this vanishing.

For $(x,y)$ in the closed cap, $|\tilde{x}|\le \epsilon$ and $|\tilde{y}|\le \sqrt{\epsilon }$. Choose $\epsilon$ so small that $\sqrt{{\epsilon }^2+\epsilon }<r_g$, with a strict margin. The whole closed cap then lies in the common real analytic solution neighborhood. Pulling the solution back provesIn fact $v$ is real analytic on a neighborhood of its closure, which is useful for the boundary integration below.

For each polynomial $g$, take the real analytic adjoint solution on the common cap from the preceding part. On the flat boundary face the full boundary jet of $u$ vanishes; on the curved face that of $v$ vanishes. The complete boundary-jet condition for formal adjoints, proved under its corrected hypothesis, therefore applies. The two corner points have zero boundary measure, and the cap is a bounded piecewise smooth domain. It follows thatBy the Weierstrass approximation theorem, polynomials uniformly approximate $\bar{u}$ on the compact cap closure. Taking the limit gives ${\int }_{{\omega }_{\epsilon }}|u{|}^2=0$. Continuity then gives $u=0$ pointwise in the cap. For real $u$, simply approximate $u$ itself.

A positive cap alone is not a neighborhood of the origin. Apply the same argument after reflecting $x\mapsto -x$; the reflected operator remains real analytic and non-characteristic. Shrinking the two parameters to their minimum proves vanishing onThis set is open and contains the origin, including the flat face where the value data vanish. The argument is the parabolic-cap proof of Holmgren uniqueness, and it establishes Holmgren uniqueness theorem for nonanalytic $C^m$ solutions here.