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

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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

Matteo Novaga ^1, , Diego Pallara ^2, and Yannick Sire ^3,

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

Abstract
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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.

Keywords: Wiener spaces., Fractional Ornstein-Uhlenbeck operator.

Mathematics Subject Classification: Primary: 35R15, 35R11, 35J61; Secondary: 35J7.

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 and 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), 9-33. doi: 10.1023/A:1010602715526.
[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.
[3]	V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.
[4]	X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differential Equations, 49 (2014), 233-269. doi: 10.1007/s00526-012-0580-6.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[6]	A. Cesaroni, M. Novaga and A. Pinamonti, One-dimensional symmetry for semilinear equations with unbounded drift. Comm. Pure Appl. Analysis, 12 (2013), 2203-2211. doi: 10.3934/cpaa.2013.12.2203.
[7]	A. Cesaroni, M. Novaga and E. Valdinoci, A symmetry result for the Ornstein-Uhlenbeck operator, Discrete Contin. Dyn. Syst.-A, 34 (2014), 2451-2467. doi: 10.3934/dcds.2014.34.2451.
[8]	E. De Giorgi, Convergence problems for functionals and operators, in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 131-188, Pitagora, Bologna, 1979. Also in: Ennio De Giorgi: Selected Papers (L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo eds.), 487-516, Springer, 2006.
[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, Contemp. Math., 528, Amer. Math. Soc., Providence, RI, (2010), 115-137. doi: 10.1090/conm/528/10418.
[10]	E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[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), 741-791.
[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, 74-96, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789812834744_0004.
[13]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.
[14]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. doi: 10.1090/S0002-9947-1972-0293384-6.
[15]	M. Novaga, D. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space,, J. Anal. Math., ().
[16]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[17]	I. Shigekawa, Stochastic Analysis, American Mathematical Society, 2004.
[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), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.
[19]	P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.
[20]	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.
[21]	P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.
[22]	K. Yosida, Functional Analysis. Sixth Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123 Springer-Verlag, Berlin-New-York, 1980.

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), 9-33. doi: 10.1023/A:1010602715526.
[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.
[3]	V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.
[4]	X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differential Equations, 49 (2014), 233-269. doi: 10.1007/s00526-012-0580-6.
[5]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[6]	A. Cesaroni, M. Novaga and A. Pinamonti, One-dimensional symmetry for semilinear equations with unbounded drift. Comm. Pure Appl. Analysis, 12 (2013), 2203-2211. doi: 10.3934/cpaa.2013.12.2203.
[7]	A. Cesaroni, M. Novaga and E. Valdinoci, A symmetry result for the Ornstein-Uhlenbeck operator, Discrete Contin. Dyn. Syst.-A, 34 (2014), 2451-2467. doi: 10.3934/dcds.2014.34.2451.
[8]	E. De Giorgi, Convergence problems for functionals and operators, in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 131-188, Pitagora, Bologna, 1979. Also in: Ennio De Giorgi: Selected Papers (L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, S. Spagnolo eds.), 487-516, Springer, 2006.
[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, Contemp. Math., 528, Amer. Math. Soc., Providence, RI, (2010), 115-137. doi: 10.1090/conm/528/10418.
[10]	E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[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), 741-791.
[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, 74-96, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789812834744_0004.
[13]	N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann., 311 (1998), 481-491. doi: 10.1007/s002080050196.
[14]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. doi: 10.1090/S0002-9947-1972-0293384-6.
[15]	M. Novaga, D. Pallara and Y. Sire, A fractional isoperimetric problem in the Wiener space,, J. Anal. Math., ().
[16]	O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2), 169 (2009), 41-78. doi: 10.4007/annals.2009.169.41.
[17]	I. Shigekawa, Stochastic Analysis, American Mathematical Society, 2004.
[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), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.
[19]	P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.
[20]	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.
[21]	P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.
[22]	K. Yosida, Functional Analysis. Sixth Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123 Springer-Verlag, Berlin-New-York, 1980.

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