| Abstract: |
| In this talk we will discuss some results for bounded positive solutions of the capillary overdetermined problem:
\[
\left\{\begin{array} {ll}
\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; \Omega \subset \mathbb{R}^n,\
u= 0 & \mbox{on }\; \partial \Omega, \
\partial_{\nu} u= const. &\mbox{on }\; \partial \Omega.
\end{array}\right.
\]
In particular, we will show rigidity for some class of domains $\Omega$, including some gradient estimates, and existence of bifurcation for other classes of domains. These are joint works with Y. Lian. |
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