| Abstract: |
| We consider overdetermined semilinear elliptic problems in bounded domains contained in an unbounded cylinder in $\mathbb{R^N}$.
The variational formulation of these problems naturally leads to study the stationary points (under a volume constraint) of a corresponding energy functional. In particular, domains which are local minima of the energy are of special interest.
We will present several results which show that the domains with the simplest geometry (namely bounded cylinders) are not always the best candidates to minimize the energy. This, in turn, suggests the existence of nontrivial domains for which the overdetermined problem admits a solution which can be obtained by globally minimizing the energy or by a bifurcation analysis. |
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