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We study nonlinear elliptic equations of the form where the main assumption on and is that there exists a one dimensional solution which solves the equation in all the directions . We show that entire monotone solutions are one dimensional if their level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.
We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional among all functions which are fixed on .
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