Publication: On global solutions to semilinear elliptic equations related to the one-phase free boundary problem
On global solutions to semilinear elliptic equations related to the one-phase free boundary problem
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Fernández-Real, X., & Ros-Oton, X. (2019). On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems. Series A, 39(12), 6945–6959. https://doi.org/10.3934/dcds.2019238
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Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of Δu=f(u) in Rn, where f is smooth, non-negative, with support in the interval [0,1]. In such setting, any "blow-down" of the solution u will converge to a global solution to the classical one-phase free boundary problem of Alt–Caffarelli. In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions u that are energy minimizers. Our main result establishes that, in dimensions n<6, if u is
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- Fernández-Real, Xavier
- Ros-Oton, Xavier
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Fernández-Real, X., & Ros-Oton, X. (2019). On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems. Series A, 39(12), 6945–6959. https://doi.org/10.3934/dcds.2019238
