On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE

Abstract

Let $L=\sum _{i,j}{a}_{ij}{\mathrm{\partial }}_{x_i}{\mathrm{\partial }}_{x_j}+\sum _i{b}_i{\mathrm{\partial }}_{x_i}+c$ be an elliptic operator with smooth enough coefficients, u a solution to the equation Lu = 0 in $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $\omega \subset \mathrm{\Omega }$ an open set. As is well known, if $u{|}_{\omega }=0$ then u = 0 everywhere in Ω. Let P be a polynomial of variables ${\tau }_1,\dots ,{\tau }_m$, functions $u_1,\dots ,u_m$ the solutions to Lu = 0; let $p\left(x\right)=P\left(u_1\right(x),\dots ,u_m(x\left)\right)$. If the coefficients of the operator are (real) analytic then $u_1,\dots ,u_m,p$ are analytic and $p{|}_{\omega }=0$ implies $p\equiv 0$ in Ω. Is the same true for smooth but not analytic coefficients? The question also concerns the polynomials of harmonic quaternion fields. In general, the answer turns out to be negative: the paper provides the relevant counterexamples.

COMMUNICATED BY:

• J. N. Wang

Keywords:

• Second-order elliptic PDE
• uniqueness of continuation of solutions
• polynomials of solutions
• uniqueness of continuation of polynomials

2010 Mathematics Subject Classifications:

• 35J15
• 58J05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the RFBR grant 18-01-00269 and Volks-Wagen Foundation.