## 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

### Keywords:

level sets; hyperplanes De Giorgi-conjecture

### Citations:

Zbl 0918.35046; Zbl 0968.35041; Zbl 1079.35513
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