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BIRS Workshop Lecture Videos

Symmetry results in the half space for a semi-linear fractional Laplace equation\\ through a one-dimensional analysis Quaas, Alexander


In this talk we analyze the semi-linear fractional Laplace equation $$(-\Delta)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\R^N_+=\{x=(x',x_N)\in \R^N:\ x_N>0\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $\rho:=\sup_{\mathbb{R}^N}u$ verifying $f(\rho)=0$, then $u$ is necessarily one-dimensional.

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