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On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. (English) Zbl 1121.35312

The paper deals with entire smooth solutions \(u:\mathbb R^n\to\mathbb R\) of the elliptic equation \(\Delta u= F'(u)\). Here \(F:\mathbb R\to\mathbb R\) is any smooth function. A generalization of the De Giorgi-conjecture states: Assume that the solution \(u\) is bounded and satisfies the monotonicity assumption \(\frac{\partial u}{\partial x_n}>0\). Then all the level sets \(\{x:u(x)=s\}\) are hyperplanes, provided that \(n\leq 8\).
This conjecture was proved by N. Ghoussoub and C. Gui [Math. Ann. 311, 481–491 (1998; Zbl 0918.35046)] in \(n=2\) and by the second and third author [J. Am. Math. Soc. 13, 725–738 (2000; Zbl 0968.35041)] in \(n=3\) for a special class of nonlinearities \(F'\). In the present paper, the De Giorgi-conjecture is proved in \(\mathbb R^3\) for any smooth \(F\).
Basing upon a stability property of the entire monotonic solution and a related local minimality result for the corresponding variational functional in \(u\), the estimate \(\int_{B_R}|\nabla u|^2\,dx\leq CR^{n-1}\) is deduced. If \(n=3\), this is enough to apply a Liouville result due to H. Berestycki, L. Caffarelli and L. Nirenberg [Ann. Sci. Norm. Sup. Pisa, Cl. Sci. (4) 25, No. 1–2, 69–94 (1998; Zbl 1079.35513)] to the auxiliary function \(\sigma_i:= \frac{\partial_iu}{\partial_nu}\). These functions are known to solve the equations \(\text{div}((\partial_nu)^2\nabla\sigma_i)=0\).
For \(n\leq 8\) an “asymptotic” version of the De Giorgi-conjecture is proved, while the original conjecture remains open even for the model nonlinearity \(F'(u)= u^3-u\), if \(4\leq n\leq 8\).

MSC:

35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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