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Karakhanyan","submitted_at":"2017-02-01T21:49:09Z","abstract_excerpt":"Let $u$ be a weak solution of the free boundary problem $$\\mathcal L u=\\lambda_0 \\mathcal H^1\\lfloor\\partial\\{u>0\\}, u\\ge 0,$$ where $\\mathcal L u={\\text{div}}(g(\\nabla u)\\nabla u)$ is a quasilinear elliptic operator and $g(\\xi)$ is a given function of $\\xi$ satisfying some structural conditions. We prove that the free boundary $\\partial\\{ u>0\\}$ is continuously differentiable in $\\mathbb R^2$, provided that $\\partial\\{ u>0\\}$ has locally finite connectivity. Moreover, we show that the free boundaries of weak solutions with finite $\\it{Morse \\ index}$ must have finite connectivity. 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