Bernstein and De Giorgi type problems: new results via a geometric approach

Farina, Alberto; Sciunzi, Berardino; Valdinoci, Enrico

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 741-791.

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form

$$\mathrm{div}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$

Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${ℝ}^2$ and ${ℝ}^3$ and of the Bernstein problem on the flatness of minimal area graphs in ${ℝ}^3$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to very degenerate operators: as an application, we prove one-dimensional symmetry for $1$-Laplacian type operators.

[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. | MR | Zbl

[2] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in ${ℝ}^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000) (electronic), 725-739. | MR | Zbl

[3] H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997-1998), 69-94. | Numdam | MR | Zbl

[4] S. Bernstein, Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Math. Z. 26 (1927), 551-558. | JFM | MR

[5] L. Caffarelli, N. Garofalo and F. Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), 1457-1473. | MR | Zbl

[6] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of $m$-Laplace equations, J. Differential Equations 206 (2004), 483-515. | MR | Zbl

[7] D. Danielli and N. Garofalo, Properties of entire solutions of non-uniformly elliptic equations arising in geometry and in phase transitions, Calc. Var. Partial Differential Equations 15 (2002), 451-491. | MR | Zbl

[8] E. De Giorgi, Convergence problems for functionals and operators, In: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978)”, Bologna, 1979, 131-188, Pitagora Editrice. | MR | Zbl

[9] E. Dibenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. | MR | Zbl

[10] L. C. Evans and R. F. Gariepy, “Measure theory and fine properties of functions”, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. | MR | Zbl

[11] A. Farina, “Propriétés qualitatives de solutions d'équations et systèmes d'équations non-linéaires”, Habilitation à diriger des recherches, Paris VI, 2002.

[12] A. Farina, One-dimensional symmetry for solutions of quasilinear equations in ${ℝ}^2$, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 6 (2003), 685-692. | MR | Zbl

[13] A. Farina, Liouville-type theorems for elliptic problems, In: “Handbook of Differential Equations: Stationary Partial Differential Equations” M. Chipot (ed.), Vol. IV Elsevier B. V., Amsterdam, 2007, 61-116. | MR | Zbl

[14] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Preliminary version of this paper, available at http://cvgmt.sns.it/papers/, 2007.

[15] A. Farina and E. Valdinoci, $1$D symmetry for solutions of semilinear and quasilinear elliptic equations, Preprint, 2008. | MR | Zbl

[16] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211. | MR | Zbl

[17] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), 481-491. | MR | Zbl

[18] D. Gilbarg and N. S. Trudinger, “Elliptic partial differential equations of second order”, Classics in Mathematics, Springer-Verlag, Berlin, 2001. | MR | Zbl

[19] E. H. Lieb and M. Loss, “Analysis”, Vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[20] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), 679-684. | MR | Zbl

[21] W. F. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), 219-226. | MR | Zbl

[22] O. Savin, Regularity of flat level sets in phase transitions, to appear in Ann. of Math. 2008. | MR | Zbl

[23] E. Sernesi, “Geometria 2”, Bollati Boringhieri, Torino, 1994.

[24] L. Simon, “Singular Sets and Asymptotics in Geometric Analysis” 2007.

[25] Y. S. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, preprint, 2008. | Zbl

[26] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal. 141 (1998), 375-400. | MR | Zbl

[27] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math. 503 (1998), 63-85. | MR | Zbl

[28] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. | MR | Zbl

[29] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 219-240. | MR | Zbl

[30] E. Valdinoci, B. Sciunzi and V. O. Savin, Flat level set regularity of $p$-Laplace phase transitions, Mem. Amer. Math. Soc. 182 (2006), vi-144. | MR | Zbl