December  2019, 39(12): 6945-6959. doi: 10.3934/dcds.2019238

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

1. 

Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

2. 

Institut für Mathematik, Universität Zürich, Winterthurerstrasse, 8057 Zürich, Switzerland

* Corresponding author

Dedicado con afecto a Luis Caffarelli, cuyos trabajos han influenciado a toda una nueva generación de matemáticos.

Received  September 2018 Revised  February 2019 Published  June 2019

Fund Project: 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.

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.

Citation: Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238
References:
[1]

H. W. Alt and L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[2]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $ \mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[3] J. D. Buckmaster and G. S. Ludford, Theory of Laminar Flames, Cambridge Univ. Press, Cambridge, 1982.   Google Scholar
[4]

X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Applied Mathematics, 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327.  Google Scholar

[5]

X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^n$, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 769-774.  doi: 10.1016/j.crma.2004.03.013.  Google Scholar

[6]

X. Cabré and X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.  doi: 10.1080/03605302.2012.697505.  Google Scholar

[7]

X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of $ \mathbb{R}^{2m}$, J. Eur. Math. Soc., 11 (2009), 819-843.  doi: 10.4171/JEMS/168.  Google Scholar

[8]

L. CaffarelliD. Jerison and C. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimension, Contemp. Math., 350 (2004), 83-97.  doi: 10.1090/conm/350/06339.  Google Scholar

[9]

L. Caffarelli and S. Salsa, A Geometric Approach To Free Boundary Problems, AMS, 2005. doi: 10.1090/gsm/068.  Google Scholar

[10]

L. Caffarelli and J. L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc., 347 (1995), 411-441.  doi: 10.1090/S0002-9947-1995-1260199-7.  Google Scholar

[11]

E. De Giorgi, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), (Pitagora, Bologna, Italy), 131–188. Google Scholar

[12]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-22.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[13]

L. Dupaigne and A. Farina, Stable solutions of $ -\Delta u = f(u) $ in $ \mathbb{R}^N $, J. Eur. Math. Soc., 12 (2010), 855-882.  doi: 10.4171/JEMS/217.  Google Scholar

[14]

A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires, Habilitation à diriger des recherches, Paris Ⅵ, 2002. Google Scholar

[15]

A. Farina and E. Valdinoci, The State of the Art for a Conjecture of De Giorgi and Related Problems, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific, 2008. doi: 10.1142/9789812834744_0004.  Google Scholar

[16]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[17]

Y. LiuK. Wang and J. Wei, Global minimizers of the Allen–Cahn equation in dimension $ n = 8$, J. Math. Pures Appl., 108 (2017), 818-840.  doi: 10.1016/j.matpur.2017.05.006.  Google Scholar

[18]

Y. Liu, K. Wang and J. Wei, On one phase free boundary problem in $ \mathbb{R}^n$, preprint, arXiv: 1705.07345, (2017). Google Scholar

[19]

A. Petrosyan and N. K. Yip, Nonuniqueness in a free boundary problem from combustion, J. Geom. Anal., 18 (2007), 1098-1126.  doi: 10.1007/s12220-008-9044-9.  Google Scholar

[20]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[21]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[22]

G. S. Weiss, A singular limit arising in combustion theory: Fine properties of the free boundary, Calc. Var. PDE, 17 (2003), 311-340.   Google Scholar

show all references

References:
[1]

H. W. Alt and L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[2]

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in $ \mathbb{R}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc., 13 (2000), 725-739.  doi: 10.1090/S0894-0347-00-00345-3.  Google Scholar

[3] J. D. Buckmaster and G. S. Ludford, Theory of Laminar Flames, Cambridge Univ. Press, Cambridge, 1982.   Google Scholar
[4]

X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Applied Mathematics, 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327.  Google Scholar

[5]

X. Cabré and A. Capella, On the stability of radial solutions of semilinear elliptic equations in all of $ \mathbb{R}^n$, C. R. Acad. Sci. Paris, Ser. I, 338 (2004), 769-774.  doi: 10.1016/j.crma.2004.03.013.  Google Scholar

[6]

X. Cabré and X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.  doi: 10.1080/03605302.2012.697505.  Google Scholar

[7]

X. Cabré and J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of $ \mathbb{R}^{2m}$, J. Eur. Math. Soc., 11 (2009), 819-843.  doi: 10.4171/JEMS/168.  Google Scholar

[8]

L. CaffarelliD. Jerison and C. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimension, Contemp. Math., 350 (2004), 83-97.  doi: 10.1090/conm/350/06339.  Google Scholar

[9]

L. Caffarelli and S. Salsa, A Geometric Approach To Free Boundary Problems, AMS, 2005. doi: 10.1090/gsm/068.  Google Scholar

[10]

L. Caffarelli and J. L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc., 347 (1995), 411-441.  doi: 10.1090/S0002-9947-1995-1260199-7.  Google Scholar

[11]

E. De Giorgi, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), (Pitagora, Bologna, Italy), 131–188. Google Scholar

[12]

D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. Reine Angew. Math., 635 (2009), 1-22.  doi: 10.1515/CRELLE.2009.074.  Google Scholar

[13]

L. Dupaigne and A. Farina, Stable solutions of $ -\Delta u = f(u) $ in $ \mathbb{R}^N $, J. Eur. Math. Soc., 12 (2010), 855-882.  doi: 10.4171/JEMS/217.  Google Scholar

[14]

A. Farina, Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires, Habilitation à diriger des recherches, Paris Ⅵ, 2002. Google Scholar

[15]

A. Farina and E. Valdinoci, The State of the Art for a Conjecture of De Giorgi and Related Problems, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific, 2008. doi: 10.1142/9789812834744_0004.  Google Scholar

[16]

D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal., 25 (2015), 1240-1257.  doi: 10.1007/s00039-015-0335-6.  Google Scholar

[17]

Y. LiuK. Wang and J. Wei, Global minimizers of the Allen–Cahn equation in dimension $ n = 8$, J. Math. Pures Appl., 108 (2017), 818-840.  doi: 10.1016/j.matpur.2017.05.006.  Google Scholar

[18]

Y. Liu, K. Wang and J. Wei, On one phase free boundary problem in $ \mathbb{R}^n$, preprint, arXiv: 1705.07345, (2017). Google Scholar

[19]

A. Petrosyan and N. K. Yip, Nonuniqueness in a free boundary problem from combustion, J. Geom. Anal., 18 (2007), 1098-1126.  doi: 10.1007/s12220-008-9044-9.  Google Scholar

[20]

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., 169 (2009), 41-78.  doi: 10.4007/annals.2009.169.41.  Google Scholar

[21]

P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.  doi: 10.1007/s002050050081.  Google Scholar

[22]

G. S. Weiss, A singular limit arising in combustion theory: Fine properties of the free boundary, Calc. Var. PDE, 17 (2003), 311-340.   Google Scholar

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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