A symmetry result for the Ornstein-Uhlenbeck operator

In 1978 E. De Giorgi formulated a conjecture concerning the one-dimensional symmetry of bounded solutions to the elliptic equation \Delta u=F'(u), which are monotone in some direction. In this paper we prove the analogous statement for the equation \Delta u -=F'(u), where the Laplacian is replaced by the Ornstein-Uhlenbeck operator. Our theorem holds without any restriction on the dimension of the ambient space, and this allows us to obtain an similar result in infinite dimensions by a limit procedure.


Introduction
A celebrated conjecture by De Giorgi [6] asks if bounded entire solutions to the equation which are strictly increasing in some direction are one-dimensional, in the sense that the level sets {u = λ} are hyperplanes, at least if n ≤ 8. This conjecture has been proved by Ghoussoub and Gui [14] in dimension n = 2, and by Ambrosio and Cabré [2] in dimension n = 3, and a counterexample has been given by del Pino, Kowalczyk and Wei in [7] for n ≥ 9. While the conjecture is still open for 4 ≤ n ≤ 8, a very nice proof has been presented by O. Savin [17] under the additional assumption that u connects −1 to 1 along the direction where it increases. See also [4] for another proof in dimension n = 2 and [12] for a review on the subject. In this paper, we are interested in a variant of (1.1) where the Laplacian ∆ is substituted by the Ornstein-Uhlenbeck operator ∆− x, ∇ . Namely, we consider the semilinear elliptic equation (1.2) ∆u − x, ∇u + f (u) = 0 and show the one-dimensional symmetry of bounded entire solutions which are monotone in some direction. Let us state our main result.
Theorem 1.1. Let n ∈ N, α ∈ (0, 1). Let u ∈ C 2 (R n ) ∩ L ∞ (R n ) be a solution of where f : R → R is a locally Lipschitz function. Assume that (1.3) ∇u(x), w > 0 for any x ∈ R n Supported by the Progetto CaRiPaRo "Nonlinear Partial Differential Equations: models, analysis, and controltheoretic problems" and the ERC grant ǫ "Elliptic Pde's and Symmetry of Interfaces and Layers for Odd Nonlinearities". 1 for some w ∈ R n . Then, u is one-dimensional, i.e. there exist U : R → R and ω ∈ R n such that u(x) = U ( ω, x ) for any x ∈ R n .
Notice that (1.2) can be regarded as the analog of (1.1) in the so-called Gauss space, that is, in R n endowed with the Gaussian instead of the Lebesgue measure. Indeed, while the Pde in (1.1) is the Euler-Lagrange equation of the Allen-Cahn Energy is the standard Gaussian probability measure. It is interesting to remark that Theorem 1.1 holds for general type of nonlinearities, as it happens for the conjecture of De Giorgi when n ≤ 3 (see [1], and this is a major difference with respect to the techniques in [17]). As in the case of the Laplacian, Theorem 1.1 is closely related to the Bernstein problem in the Gauss space, which asks for flatness of entire minimal surfaces which are graphs in some direction. We point out that minimal surfaces in the Gauss space are interesting geometric objects, since they correspond to self-similar shrinkers of the mean curvature flow (see for instance [8]), and satisfy the equation where κ is the mean curvature at x and ν is the normal vector. In this context, the analog of the Bernstein Theorem has been proved by Ecker and Huisken [8], under a polynomial growth assumption on the volume of the minimal surface, and more recently by Wang in [20] without any further assumption. We point out that, differently from the Euclidean case, the result holds without any restriction on the dimension of the ambient space, and in fact there is no such restriction also in Theorem 1.1. This is due to the exponential decay of the Gaussian measure associated to the Ornstein-Uhlenbeck operator which allows for better estimates than the corresponding Euclidean ones. Since Theorem 1.1 holds in any dimension and the Gauss space (R n , γ) formally converges to a Wiener space (X, H, γ) (see Section 2.1 for a precise definition) as n → ∞, one may expect that an analogous result holds in such infinite dimensional setting. Indeed, in this paper we confirm this expectation and show the infinite dimensional extension of Theorem 1.1: where f : R → R is a locally Lipschitz function. Assume that for all x ∈ X, for all R > 0 and for some w ∈ H. Then, u is one-dimensional, in the sense that there exist U : R → R and ω ∈ X * such that Notice that Theorem 1.1 can be recovered as a corollary of Theorem 1.2, when the function u depends only on finitely many variables. As far as we know, Theorem 1.1 is the first result of De Giorgi conjecture type in an infinite dimensional setting. The proof that we perform exploits and generalizes some geometric ideas of [18,19,10,11].

Notation
We denote by (R n , γ) the n-dimensional Gauss space, where γ is the standard Gaussian measure on R n defined in (1.6).
2.1. The Wiener space. An abstract Wiener space is defined as a triple (X, γ, H) where X is a separable Banach space, endowed with the norm · X , γ is a nondegenerate centered Gaussian measure, and H is the Cameron-Martin space associated to the measure γ, that is, H is a separable Hilbert space densely embedded in X, endowed with the inner product [·, ·] H and with the norm | · | H . The requirement that γ is a centered Gaussian measure means that for any x * ∈ X * , the measure x * # γ is a centered Gaussian measure on the real line R, that is, the Fourier transform of γ is given bŷ here the operator Q ∈ L(X * , X) is the covariance operator and it is uniquely determined by formula The nondegeneracy of γ implies that Q is positive definite: the boundedness of Q follows by Fernique's Theorem (see for instance [5, Theorem 2.8.5]), asserting that there exists a positive number β > 0 such that This implies also that the maps x → x, x * belong to L p γ (X) for any x * ∈ X * and p ∈ [1, +∞), where L p γ (X) denotes the space of all functions f : In particular, any element x * ∈ X * can be seen as a map x * ∈ L 2 γ (X), and we denote by R * : X * → H the identification map R * x * (x) := x, x * . The space H given by the closure of R * X * in L 2 γ (X) is called reproducing kernel. By considering the map R : H → X defined as we obtain that R is an injective γ-Radonifying operator, which is Hilbert-Schmidt when X is Hilbert. We also have Q = RR * : X * → X. The space H := RH, equipped with the inner product [·, ·] H and norm | · | H induced by H via R, is the Cameron-Martin space and is a dense subspace of X. The continuity of R implies that the embedding of H in X is continuous, that is, there exists c > 0 such that We have also that the measure γ is absolutely continuous with respect to translation along is absolutely continuous with respect to γ with density given by

Cylindrical functions and differential operators.
For j ∈ N we choose x * j ∈ X * in such a way thatĥ j := R * x * j , or equivalently h j := Rĥ j = Qx * j , form an orthonormal basis of H. We order the vectors x * j in such a way that the numbers The map Π m induces the decomposition X ≃ H m ⊕ X ⊥ m , with X ⊥ m := ker(Π m ), and γ = γ m ⊗ γ ⊥ m , with γ m and γ ⊥ m Gaussian measures on H m and X ⊥ m respectively, having H m and H ⊥ m as Cameron-Martin spaces. When no confusion is possible we identify H m with R m ; with this identification the measure γ m = Π m# γ is the standard Gaussian measure on R m (see [5]). Given x ∈ X, we denote by x m ∈ H m the projection Π m (x), and by , with continuous and bounded derivatives up to the order k. We denote by FC k b (X, H) the space generated by all functions of the form uh, with u ∈ FC k b (X) and h ∈ H. We let where ∂ j := ∂ h j and ∂ * j := ∂ j −ĥ j is the adjoint operator of ∂ j . With this notation, the integration by parts formula holds: In particular, thanks to (2.2), the operator ∇ γ is closable in L p γ (X), and we denote by W 1,p γ (X) the domain of its closure. The Sobolev spaces W k,p γ (X), with k ∈ N and p ∈ [1, +∞], can be defined analogously [5], and FC k b (X) is dense in W j,p γ (X), for all p < +∞ and k, j ∈ N with k ≥ j.
Given a vector field ϕ ∈ L p γ (X, H), p ∈ (1, ∞], using (2.2) we can define div γ ϕ in the distributional sense, taking test functions u in W 1,q γ (X) with 1 p + 1 q = 1. We say that div γ ϕ ∈ L p γ (X) if this linear functional can be extended to all test functions u ∈ L q γ (X). This is true in particular which, summing up in j, gives The operator ∆ γ : W 2,p γ (X) → L p γ (X) is usually called the Ornstein-Uhlenbeck operator. Notice that, if u is a cylindrical function, that is u(x) = v(y) with y = Π m (x) ∈ R m and m ∈ N, then We write u ∈ C(X) if u : X → R is continuous and u ∈ C 1 (X) if both u : X → R and ∇ γ u : X → H are continuous.
For simplicity of notation, from now on we will omit the explicit dependence on γ of operators and spaces. We also indicate by [·, ·] and | · | respectively the scalar product and the norm in H. When no confusion is possible, we shall also write u i to indicate the derivative ∂ i u.  which is meaningful for u ∈ W 1,2 (X). Notice that, as FC 1 b (X) is dense in W 1,2 (X), it is enough to require (3.1) for all ϕ ∈ FC 1 b (X).
3.1. The linearized equation. We now consider the equation solved by the derivatives of the solution u.
Proof. Notice first that it is enough to prove (3.2) for all ϕ ∈ FC 2 b (X). Letting ϕ ∈ FC 2 b (X), we multiply (1.8) by ϕ i and recall (2.4), to get where the last inequality follows from (3.1), with ϕ replaced by x * i , x ϕ.

3.2.
A variational inequality implied by the monotonicity. The next result shows that monotone solutions of (1.8) satisfy a variational inequality. In the Euclidean case, this fact boils down to the classical stability condition (namely, the second derivative of the energy functional being nonnegative). Differently from this, in our case, a negative eigenvalue appears in the inequality.

A geometric Poincaré inequality.
We show that a sort of geometric Poincaré inequality stems from solutions of (1.8) satisfying (3.3). In the Euclidean case, it boils down to the inequality discovered in [18,19].
Proof. We use (3.3) with test function |∇u| ϕ, and we see that (3.5) We now exploit (3.2) with test function u i ϕ 2 and we get Summing over i ∈ N, we conclude that (3.6) From (3.5) and (3.6), we conclude that which gives (3.4).
Let u ∈ C 1 (X) ∩ L ∞ (X) satisfying (1.9), let N ∈ N and x N ∈ X ⊥ N . We consider the map ψ N, be its noncritical set. By the Implicit Function Theorem, the level set of ψ N,x N in N N (x N ) are (N − 1)-dimensional hypersurfaces of class C 2 . Thus we can consider the principal curvatures of these hypersurfaces, that we denote by κ 1,N , . . . , κ N −1,N , and the tangential gradient of ψ N,x N 1 , that we denote by ∇ T,N . We also set With this notation, we have the following (3.7) 1 the tangential gradient of a function g along a hypersurface with normal ν is ∇g − (∇g · ν)ν, that is, the tangential component of the full gradient Proof. Let Since |∇ N −1 u| ≤ |∇ N u| and for any N ∈ N. Moreover, by Stampacchia's Theorem we have that ∇ N |∇ N u| = 0 for almost any x N ∈ R N \ N N (x N ), and similarly u ij = 0 for almost any x N ∈ R N \ N N (x N ). Therefore On the other hand, by [19, Formula (2.1)], From this, (3.8) and (3.9), we obtain which, recalling (3.4), implies (3.7).

A symmetry result.
We now use the previous material to obtain a one-dimensional symmetry result for the N -dimensional projection of the solution. The idea of using geometric Poincaré inequalities as the ones in [18,19] in order to obtain symmetry properties goes back to [10] and it was widely used in [11] in the finite dimensional Euclidean setting. The result we present here is the following: (1.8), (1.9) and (3.3). Then, the map ψ N,x N is one-dimensional, i.e. there exists U N,x N : R → R and ω N, Proof. We fix R > 1, to be taken arbitrarily large in what follows, and let Λ = max i λ i .
Therefore, by sending R → +∞ in (3.11), we conclude that for any x ∈ N N . From this and [11, Lemma 2.11] we get (3.10).
From the finite dimensional symmetry result of Proposition 3.6, one can take the limit as N → +∞ and obtain: Corollary 3.7. Let u ∈ C 1 (X) ∩ L ∞ (X) satisfy (1.8), (1.9) and (3.3). Then, u is necessarily one-dimensional, i.e. there exists U : R → R and ω ∈ X * such that for any x ∈ X.
Proof. We first show that there exists h ∈ H such that Let V ⊂ X be defined as V = ∪ N H N . Since V is a dense subset of X, it is enough to show that (3.12) However, from Proposition 3.6 we know that (3.13) From (3.12) it follows that there exists a function U : R → R such that U (t) = u(th) for all t ∈ R, and (3.14) Moreover, U is a bounded nondecreasing solution to the ODE Being u continuous, if U is nonconstant (otherwise the thesis follows immediately) then the functionĥ is also continuous, so that h ∈ QX * andĥ(x) = ω, x for some ω ∈ X * , which implies the thesis.

Heteroclinic solutions
The results in Theorems 1.1 and 1.2 may be seen either as classification results (when one knows explicitly the solutions of the associated one-dimensional problem) or as nonexistence result (when the associated one-dimensional problem does not admit any solution). For this, we now give some simple conditions on the nonlinearity f ensuring existence or nonexistence of bounded solutions to the ODE Notice that, from (4.2) it follows that there exist U ± ∈ R, with U − < U + , such that Moreover, passing to the limit in (4.1) we also get We start with a nonexistence result.
Proof. Let us assume that f ≤ 0 in [U 0 , U + ], since the argument is analogous in the other case, and assume by contradiction that we are given a solution U of (4.1), (4.2). Letting t 0 > 0 be such that u(t 0 ) ∈ [U 0 , U + ], we have that U satisfies the differential inequality U ′′ ≥ t U ′ for all t ∈ [t 0 , +∞), which implies, by direct integration,
As a counterpart of the nonexistence result in Proposition 4.1, we now give an existence result for monotone solutions to (4.1).
Then, there exists a monotone solution to (4.1), connecting −c to c.
Let now U ⋆ be the Ehrhard rearrangement of U [9], which is defined in such a way that U ⋆ is nondecreasing on (0, +∞), and γ t : U ⋆ (t) > r = γ t : U (t) > r for all r ∈ (0, c).
In particular, we may assume that U = U ⋆ , i.e. that U is nondecreasing on (0, +∞).
As U = c and U = 0 are solutions to (4.1), which is the Euler-Lagrange equation of G, we get that either U = 0 or U = c or (4.9) U (t) ∈ (0, c) for all t ∈ (0, +∞).
On the other hand, thanks to (4.7) and the fact that U (0) = 0, we can exclude the first two possibilities, so that (4.9) holds. Moreover, since U is nondecreasing and f (r) = 0 for all r ∈ (0, c), it follows that U ′ (t) > 0 for all t ∈ (0, +∞) and lim t→+∞ U (t) = c.
Since by (4.6) the function t → −U (−t) is a monotone solution to (4.1) on (−∞, 0), we get that the odd extension of U on R is a solution to (4.1) on the whole of R which satisfies (4.2) and connects −c to c. In particular, condition (4.7) is verified whenever c 0 2F (r) dr ≤ π 2 F (0) which is satisfied, for instance, by the standard double-well potential F (t) = (1 − t 2 ) 2 /4.