### A Liouville type theorem for $p$-harmonic functions on minimal submanifolds in ${\mathbb{R}}^{n+m}$

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form$$\phantom{\rule{0.166667em}{0ex}}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in ${\mathbb{R}}^{3}$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation $${u}_{t}={\Delta}_{V}u+aulogu+bu$$ where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0,\infty )$. We suppose that the Bakry-Émery curvature and the $N$-dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation ${u}_{t}=\Delta u+{\left|u\right|}^{p-1}u$. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence of steady states and time-periodic solutions are also shown.

We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem. The crucial ingredients of our proofs are the understanding of the blow up profiles and a Liouville theorem.

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi's theorem, Beurling's theorem, and Donoho-Stark's uncertainty principle are obtained for the Weinstein transform.