Classification of solutions of an elliptic Hamilton-Jacobi equation

TL;DR

This work resolves the open question of whether classical solutions to the elliptic Hamilton-Jacobi equation $-\Delta u = |\nabla u|^p$ in a half-space are necessarily one-dimensional when $1<p\le 2$, under zero Dirichlet boundary conditions. By establishing a global boundedness result through a rescaling-oscillation barrier framework, the authors enable the use of moving planes/translation-compactness arguments to deduce that any solution depends only on the normal variable $x_n$. The $p=2$ case is treated via the Hopf–Cole transformation to harmonic functions, completing the full classification for all $p>1$. The approach removes prior boundedness assumptions and extends the rigidity phenomenon to the full range, with implications for gradient blow-up analysis and Liouville-type results in nonlinear elliptic PDEs.

Abstract

We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-Δu= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known result, which required an extra assumption of boundedness from above. Combined with the existing analogous result for $p>2$, our result completes the full classification picture of the Dirichlet problem for equation (DHJ) in a half-space.

Theorems & Definitions (13)

• Remark 1.1
• Theorem 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5