Thursday, October 24, 2024 — 11:00 am — aula M2.2, edificio Matematica
Liouville type theorems for anisotropic degenerate elliptic equations on strips
Abstract: We discuss some recent results concerning ($L^\infty$) Liouville type theorems for anisotropic degenerate elliptic equations in divergence form on the strip $S={\mathbb R}^{N-1}\times (-1,1)$ where $x=(x’,\lambda)$. The model equation is $div_{x’} (w_1 \nabla_{x’}\sigma)+\partial_\lambda (w_1 w_2 \partial_\lambda \sigma)=0$, where $w_i(x’,\lambda)$ are positive and locally bounded in $S$.
We deduce them by means of a modification of De Giorgi’s oscillation decrease argument for uniformly elliptic equations, under appropriate conditions on the weight functions $w_i$; the key one being the existence of a positive unbounded supersolution close to the degeneration set $\partial S$.
For example our approach works in the case $w_1=1-|\lambda|$ and $w_2=(1-|\lambda|)^2$, for which the corresponding ($L^\infty$) Liouville type theorem entails an alternative proof of the (known) positive answer to a famous conjecture of De Giorgi in any space dimension under the additional assumption that the zero level set of the solution is a Lipschitz graph; moreover it is related to an ($L^\infty$) Liouville type theorem for an isotropic degenerate elliptic equation on $\mathbb R^N$ with a one-dimensional weight decaying exponentially at infinity in one direction.
A complete picture of the problem is given for weights $w_1=(1-|\lambda|)^{\alpha}$, with $\alpha>-1$ and $w_2=(1-|\lambda|)^{\nu}$. The case $\nu=2$ and $\nu=1-\alpha$ being borderline cases.
When $\nu<1-\alpha$, a bounded non constant solution exists, this case is connected to fractional Laplacians with fractional order $\frac{\alpha+1}{2-\nu}$.
On the other hand, when $1-\alpha\le \nu<2$, (one sided) Liouville type results are proved, that is any nonnegative solution is indeed constant; this case is strictly related to the Laplace operator on ${\mathbb R}^{N-1}\times B(0,1)$, where $B(0,1)\subset {\mathbb R}^{\frac{2(\alpha+1)}{2-\nu}}$, when this exponent is integer, as it is the case for $\nu=1-\alpha$.
Finally, in the so called supercritical case, $\nu>2$, as well as in the critical case, $\nu=2$, ($L^\infty$) Liouville type theorems are proved but one-sided Liouville type theorems need not to be true.
The talk is mainly based on the following works:
- “Liouville type theorems for anisotropic degenerate elliptic equations on strips”, L. Moschini, CPAA 2023,
- “Anisotropic degenerate elliptic operators with distance function weights on strips”, S. Filippas, L. Moschini and A. Tertikas (submitted).