Research Problems in Function Theory — Problem 7.49

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $n\ge3$, an integer $m\ge2$ and $p\in(1,\infty)\setminus\{2\}$. A real polynomial $u$ is homogeneous of degree $m$ if $u(tx)=t^m u(x)$ for every $t>0$. Write $Du$ for its gradient, $D^2u$ for its Hessian and $\Delta u$ for the sum of its second partial derivatives. The polynomial form of the $p$-Laplace equation is \[|Du|^2\Delta u+(p-2)\langle D^2u\,Du,Du\rangle=0.\] At points where $Du\ne0$ this is equivalent to $\operatorname{div}(|Du|^{p-2}Du)=0$; the displayed polynomial identity also makes sense at critical points.

Statement recorded in the supplied source.

Must every real homogeneous polynomial of degree $m\ge2$ satisfying this identity throughout $\mathbb R^n$ be identically zero, for every $n\ge3$ and every $1<p<\infty$, $p\ne2$? [S2] Tkachev's Conjecture1.1 explicitly states Lewis's full question and fixes the intended differential equation, rather than the malformed gradient-dot expression printed in the problem list. The paper records the quadratic case and proves the cubic case without restricting the degree of the question. [S3]

Short English statement

Can a nonzero real homogeneous polynomial of degree at least two solve the nonquadratic p-Laplace equation in dimension at least three?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307049 (AMR-022-7049). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.49 and its complete update. https://arxiv.org/abs/1809.07200.
• S3 Vladimir G. Tkachev, On the non-vanishing property for real analytic solutions of the $p$-Laplace equation (2015), Section1, equation(1.1), Conjecture1.1 and Theorem1.2, pages1--2. https://arxiv.org/abs/1503.03234.