Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 105 / 1 / 2 / h

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.