| Abstract: |
| In this talk, we discuss superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional
\[
\int_{\Omega}\left(|\nabla\mathbf{u}|^2+\frac{2}{p}|\mathbf{u}|^p\right),\quad 0\lt p\lt 1,
\]
but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,\alpha}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity. |
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