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June  2016, 9(3): 815-831. doi: 10.3934/dcdss.2016030

## A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications

 1 Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa 2 Dipartimento di Matematica e Fisica "Ennio De Giorgi" & INFN, Università del Salento, P.O.B. 193, 73100, Lecce, Italy 3 Université Aix-Marseille, I2M, UMR CNRS 7353, Marseille, France

Received  November 2014 Revised  April 2015 Published  April 2016

The purpose of this paper is to study a boundary reaction problem on the space $X \times {\mathbb R}$, where $X$ is an abstract Wiener space. We prove that smooth bounded solutions enjoy a symmetry property, i.e., are one-dimensional in a suitable sense. As a corollary of our result, we obtain a symmetry property for some solutions of the following equation $$(-\Delta_\gamma)^s u= f(u),$$ with $s\in (0,1)$, where $(-\Delta_\gamma)^s$ denotes a fractional power of the Ornstein-Uhlenbeck operator, and we prove that for any $s \in (0,1)$ monotone solutions are one-dimensional.
Citation: Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030
##### References:
 [1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property,, Acta Appl. Math., 65 (2001), 9. doi: 10.1023/A:1010602715526. 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. doi: 10.1090/S0894-0347-00-00345-3. Google Scholar [3] V. I. Bogachev, Gaussian Measures,, Mathematical Surveys and Monographs, 62 (1998). doi: 10.1090/surv/062. Google Scholar [4] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations,, Calc. Var. Partial Differential Equations, 49 (2014), 233. doi: 10.1007/s00526-012-0580-6. Google Scholar [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [6] A. Cesaroni, M. Novaga and A. Pinamonti, One-dimensional symmetry for semilinear equations with unbounded drift., Comm. Pure Appl. Analysis, 12 (2013), 2203. doi: 10.3934/cpaa.2013.12.2203. Google Scholar [7] A. Cesaroni, M. Novaga and E. Valdinoci, A symmetry result for the Ornstein-Uhlenbeck operator,, Discrete Contin. Dyn. Syst.-A, 34 (2014), 2451. doi: 10.3934/dcds.2014.34.2451. Google Scholar [8] E. De Giorgi, Convergence problems for functionals and operators,, in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1978), 131. Google Scholar [9] M. del Pino, M. Kowalczyk and J. Wei, On a conjecture by De Giorgi in dimensions 9 and higher,, in Symmetry for elliptic PDEs, 528 (2010), 115. doi: 10.1090/conm/528/10418. Google Scholar [10] E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218. Google Scholar [11] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741. Google Scholar [12] A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems,, in Recent progress on reaction-diffusion systems and viscosity solutions, (2009), 74. doi: 10.1142/9789812834744_0004. Google Scholar [13] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196. Google Scholar [14] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207. doi: 10.1090/S0002-9947-1972-0293384-6. Google Scholar [15] M. Novaga, D. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space,, J. Anal. Math., (). Google Scholar [16] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math. (2), 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [17] I. Shigekawa, Stochastic Analysis,, American Mathematical Society, (2004). Google Scholar [18] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result,, J. Funct. Anal., 256 (2009), 1842. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [19] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces,, J. Reine Angew. Math., 503 (1998), 63. Google Scholar [20] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar [21] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092. doi: 10.1080/03605301003735680. Google Scholar [22] K. Yosida, Functional Analysis. Sixth Edition,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980). Google Scholar

show all references

##### References:
 [1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property,, Acta Appl. Math., 65 (2001), 9. doi: 10.1023/A:1010602715526. 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. doi: 10.1090/S0894-0347-00-00345-3. Google Scholar [3] V. I. Bogachev, Gaussian Measures,, Mathematical Surveys and Monographs, 62 (1998). doi: 10.1090/surv/062. Google Scholar [4] X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations,, Calc. Var. Partial Differential Equations, 49 (2014), 233. doi: 10.1007/s00526-012-0580-6. Google Scholar [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [6] A. Cesaroni, M. Novaga and A. Pinamonti, One-dimensional symmetry for semilinear equations with unbounded drift., Comm. Pure Appl. Analysis, 12 (2013), 2203. doi: 10.3934/cpaa.2013.12.2203. Google Scholar [7] A. Cesaroni, M. Novaga and E. Valdinoci, A symmetry result for the Ornstein-Uhlenbeck operator,, Discrete Contin. Dyn. Syst.-A, 34 (2014), 2451. doi: 10.3934/dcds.2014.34.2451. Google Scholar [8] E. De Giorgi, Convergence problems for functionals and operators,, in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, (1978), 131. Google Scholar [9] M. del Pino, M. Kowalczyk and J. Wei, On a conjecture by De Giorgi in dimensions 9 and higher,, in Symmetry for elliptic PDEs, 528 (2010), 115. doi: 10.1090/conm/528/10418. Google Scholar [10] E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218. Google Scholar [11] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741. Google Scholar [12] A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems,, in Recent progress on reaction-diffusion systems and viscosity solutions, (2009), 74. doi: 10.1142/9789812834744_0004. Google Scholar [13] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems,, Math. Ann., 311 (1998), 481. doi: 10.1007/s002080050196. Google Scholar [14] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, Trans. Amer. Math. Soc., 165 (1972), 207. doi: 10.1090/S0002-9947-1972-0293384-6. Google Scholar [15] M. Novaga, D. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space,, J. Anal. Math., (). Google Scholar [16] O. Savin, Regularity of flat level sets in phase transitions,, Ann. of Math. (2), 169 (2009), 41. doi: 10.4007/annals.2009.169.41. Google Scholar [17] I. Shigekawa, Stochastic Analysis,, American Mathematical Society, (2004). Google Scholar [18] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result,, J. Funct. Anal., 256 (2009), 1842. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [19] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces,, J. Reine Angew. Math., 503 (1998), 63. Google Scholar [20] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains,, Arch. Rational Mech. Anal., 141 (1998), 375. doi: 10.1007/s002050050081. Google Scholar [21] P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators,, Comm. Partial Differential Equations, 35 (2010), 2092. doi: 10.1080/03605301003735680. Google Scholar [22] K. Yosida, Functional Analysis. Sixth Edition,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1980). Google Scholar
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