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On global solutions to semilinear elliptic equations related to the one-phase free boundary problem

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This work has received funding from the European Research Council (ERC) under the Grant Agreements No 721675 and No 801867. In addition, the second author was supported by the Swiss National Science Foundation and by MINECO grant MTM2017-84214-C2-1-P

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  • Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $ \Delta u = f(u) $ in $ \mathbb{R}^n $, 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 axially symmetric and stable then it is 1D.

    Mathematics Subject Classification: Primary: 35R35, 35J91; Secondary: 35B07.


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  • Figure 1.  Representation of $ \Phi_\varepsilon(t) = \int_0^t \beta_\varepsilon(s)\, ds $

    Figure 2.  Representation of the cases (ⅰ) $ a > 1 $, (ⅱ) $ a = 1 $, and (ⅲ) $ a < 1 $

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