| 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 $div_{x`} (w_1 \nabla_{x`}\sigma)+\partial_\lambda (w_1w_2
\partial_\lambda \sigma)=0$, 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.
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. For some values of $\alpha, \nu$ these operators are connected to fractional Laplacians.
The talk is mainly based on the following works: Liouville type theorems for anisotropic degenerate elliptic equations on strips, L. Moschini, CPAA 2023 and Anisotropic degenerate elliptic operators with distance function weights on strips, S. Filippas, L. Moschini and A. Tertikas (submitted). |
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