On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

TL;DR

This paper proves a De Giorgi-type conjecture for boundary reactions involving the half-Laplacian in dimension 4, showing that all bounded stable solutions are one-dimensional profiles, extending the understanding of stability and symmetry in nonlocal PDEs.

Contribution

It establishes the De Giorgi conjecture for boundary reactions with the half-Laplacian in dimension 4, demonstrating that stable solutions are necessarily one-dimensional.

Findings

All bounded stable solutions are one-dimensional profiles.

The result confirms the De Giorgi conjecture for half-Laplacian boundary reactions in dimension 4.

Analogous to stable minimal surfaces being planes in $\

Abstract

We prove that every bounded stable solution of \[ (-\Delta)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= \phi(e\cdot x)$ for some $e\in \mathbb S^2$, where $\phi:\mathbb R\to \mathbb R$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\mathbb R^{d+1}_+=\mathbb R^{d+1}\cap \{x_{d+1}\geq 0\}$ when $d = 4$, by proving that all critical points of $$ \int_{\{x_{d+1\geq 0}\}} \frac12 |\nabla U|^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} \frac 1 4 (1-U^2)^2 \,dx $$ that are monotone in $\mathbb R^d$ (that is, up to a rotation, $\partial_{x_d} U>0$) are one dimensional. Our result is analogue to the fact that stable embedded minimal surfaces in $\mathbb R^3$ are planes. Note that the corresponding result about stable solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still open.

Equations

$$(-\Delta {)}^{1/2}u+f(u)=0\text{mbox}in{\mathbb{R}}^3$$

$${\int }_{\{x_{d+1\ge 0}\}}\frac{1}{2}|\nabla U{|}^{2}dxd{x}_{d+1}+{\int }_{\{x_{d+1}=0\}}\frac{1}{4}(1-U^2{)}^{2}dx$$

$$-\Delta u=u-u^3,|u|\le 1,$$

$$\mathcal{E}_{1}(v):=\int_{\mathbb{R}^{d}}\bigg{(}\,\frac{1}{2}|\nabla v|^{2}+\frac{1}{4}(1-v^{2})^{2}\bigg{)}dx.$$

$$\mathcal{E}_{1,\varepsilon}(v):=\frac{1}{\varepsilon}\int_{\mathbb{R}^{d}}\bigg{(}\,\frac{\varepsilon^{2}}{2}|\nabla v|^{2}+\frac{1}{4}(1-v^{2})^{2}\bigg{)}dx.$$

$$u_{{\epsilon }_k}\to {\chi }_E-{\chi }_{E^c}\text{mbox}in{L}_{\mathrm{loc}}^1,$$

$$\{u_{\epsilon }=\lambda \}=\epsilon \{u=\lambda \},$$

$$\underset{x_d\to \pm \infty }{\lim }u=\pm 1.$$

$$u^{\pm }:=\underset{x_d\to \pm \infty }{\lim }u,$$

$$\text{stable solutions to eqrefclassic in Rd−1 are 1D}\Rightarrow \text{De Giorgi conjecture holds in Rd.}$$

$$\mathcal{J}(V):={\int }_{{\mathbb{R}}_{+}^{d+1}}\frac{1}{2}|\nabla V{|}^{2}dxd{x}_{d+1}+{\int }_{\{x_{d+1}=0\}}F(V)dx,$$

$$\left\{\begin{array}[]{ll}\Delta U=0&\text{in }\mathbb{R}^{d+1}_{+},\\ \partial_{\nu}U=-f(U)&\text{on }\{x_{d+1}=0\},\end{array}\right.$$

$${\int }_{{\mathbb{R}}_{+}^{d+1}}|\nabla V{|}^{2}d{x}_1\cdots d{x}_{d}d{x}_{d+1}={\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{|v(x)-v(y){|}^2}{|x-y{|}^{d+1}}dxdy$$

$$\mathcal{E}(v):={\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{1}{2}\frac{|v(x)-v(y){|}^2}{|x-y{|}^{d+1}}dxdy+{\int }_{{\mathbb{R}}^d}F(v(x))dx$$

$$\text{$U$ is a local min. of $\mathcal{J}$ in $\mathbb{R}^{d+1}_{+}$}\quad\Leftrightarrow\quad\left\{\begin{array}[]{l}\text{$u=U|_{x_{d+1}=0}$ is a local min. of $\mathcal{E}$ in $\mathbb{R}^{d}$}\\ \text{and $U$ is the harmonic extension of $u$.}\end{array}\right.$$

$$(-\Delta {)}^{1/2}u+f(u)=0\text{mbox}in{\mathbb{R}}^d,$$

$$(-\Delta {)}^{1/2}u(x)=\text{p.v.}{\int }_{{\mathbb{R}}^d}\frac{u(x)-u(y)}{|x-y{|}^{d+1}}dy.$$

$$\mathcal{E}_{\varepsilon}(v):=\frac{1}{\varepsilon\log(1/\varepsilon)}\bigg{(}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{\varepsilon}{2}\frac{|v(x)-v(y)|^{2}}{|x-y|^{d+1}}\,dx\,dy+\int_{\mathbb{R}^{d}}\frac{1}{4}(1-v^{2})^{2}\,dx\bigg{)}.$$

$$u_{{\epsilon }_k}\to {\chi }_E-{\chi }_{E^c}\text{mbox}in{L}_{\mathrm{loc}}^1,$$

$$(-\Delta {)}^{1/2}u=u-u^3,|u|\le 1,$$

$$(-\Delta {)}^{1/2}u+f(u)=0,|u|\le 1\text{mbox}in{B}_R\subset {\mathbb{R}}^d,$$

$${\int }_{B_{R/2}}|\nabla u|\le C{R}^{d-1}\log (M_{o}R)$$

$${\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d\setminus B_{R/2}^c\times B_{R/2}^c}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{d+1}}dxdy\le C{R}^{d-1}{\log }^2(M_{o}R).$$

$$\mathcal{E}(v;B_R):={\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d\setminus B_R^c\times B_R^c}\frac{1}{2}\frac{|v(x)-v(y){|}^2}{|x-y{|}^{d+1}}dxdy+{\int }_{B_R}F(v(x))dx,$$

$$\int_{B_{R}}\big{(}(-\Delta)^{1/2}\xi+f^{\prime}(u)\xi\big{)}\xi\,dx\geq 0\quad\mbox{for all }\xi\in C^{2}_{c}(B_{R}).$$

$${\phi }^0(x):=\{\begin{array}{ll}1& \text{mbox}for|x|\le \frac{1}{2}\\ 2-2|x|& \text{mbox}for\frac{1}{2}\le |x|\le 1\\ 0& \text{mbox}for|x|\ge 1,\end{array}$$

$${\phi }^1(x):=\{\begin{array}{ll}1& \text{mbox}for|x|\le \sqrt{R}\\ 2-2\frac{\log |x|}{\log R}& \text{mbox}for\sqrt{R}\le |x|\le R\\ 0& \text{mbox}for|x|\ge R,\end{array}$$

$${\phi }^2(x):=\{\begin{array}{ll}1& \text{mbox}for|x|\le R_{\ast }\\ 1-\frac{\log \log |x|}{\log \log (R)}& \text{mbox}for{R}_{\ast }\le |x|\le R\\ 0& \text{mbox}for|x|\ge R,\end{array}$$

$${\Psi }_{t,\mathbf{v}}^i(z):=z+t{\phi }^i(z)\mathbf{v},t\in \mathbb{R},z\in {\mathbb{R}}^d,i=0,1,2.$$

$$\mathcal{P}^{i}_{t,\boldsymbol{v}}v(x):=v\bigl{(}(\Psi_{t,\boldsymbol{v}}^{i})^{-1}(x)\bigr{)}.$$

Full text

[TABLE]

is a 1D profile, i.e., $u(x)=\phi(e\cdot x)$ for some $e\in\mathbb{S}^{2}$, where $\phi:\mathbb{R}\to\mathbb{R}$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\mathbb{R}^{d+1}_{+}=\mathbb{R}^{d+1}\cap\{x_{d+1}\geq 0\}$ when $d=4$, by proving that all critical points of

[TABLE]

that are monotone in $\mathbb{R}^{d}$ (that is, up to a rotation, $\partial_{x_{d}}U>0$) are one dimensional.

Our result is analogue to the fact that stable embedded minimal surfaces in $\mathbb{R}^{3}$ are planes. Note that the corresponding result about stable solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still open.

1. introduction

1.1. De Giorgi conjecture on the Allen-Cahn equation

In 1978, De Giorgi stated the following famous conjecture [14]:

Conjecture 1.1**.**

Let $u\in C^{2}(\mathbb{R}^{d})$ be a solution of the Allen-Cahn equation

[TABLE]

satisfying $\partial_{x_{d}}u>0$. Then, if $d\leq 8$, all level sets $\{u=\lambda\}$ of $u$ must be hyperplanes.

To motivate this conjecture, we need to explain its relation to minimal surfaces.

1.2. Allen-Cahn vs. minimal surfaces

It is well-known that (1.1) is the condition of vanishing first variation for the Ginzburg-Landau energy

[TABLE]

By scaling, if $u$ is a local minimizer of $\mathcal{E}_{1}$, then $u_{\varepsilon}(x):=u(\varepsilon^{-1}x)$ are local minimizers of the $\varepsilon$-energy

[TABLE]

In [23, 24], Modica and Mortola established the $\Gamma$-convergence of $\mathcal{E}_{1,\varepsilon}$ to the perimeter functional as $\varepsilon\downarrow 0$. As a consequence, the rescalings $u_{\varepsilon}$ have a subsequence $u_{\varepsilon_{k}}$ such that

[TABLE]

and $E$ is a local minimizer of the perimeter in $\mathbb{R}^{d}$. This result was later improved by Caffarelli and Cordoba [9], who showed a density estimate for minimizers of $\mathcal{E}_{1,\varepsilon}$, and proved that the super-level sets $\{u_{\varepsilon_{k}}\geq\lambda\}$ converge locally uniformly (in the sense of Hausdorff distance) to $E$ for each fixed $\lambda\in(-1,1)$. Hence, at least heuristically, minimizers of $\mathcal{E}_{1,\varepsilon}$ for $\varepsilon$ small should behave similarly to sets of minimal perimeter.

1.3. Classifications of entire minimal surfaces and De Giorgi conjecture

Here we recall some well-known facts on minimal surfaces:111Note that, here and in the sequel, the terminology “minimal surface” denotes a critical point of the area functional (in other words, a surface with zero mean curvature).

1. (i)

If $E$ is a local minimizer of the perimeter in $\mathbb{R}^{d}$ with $d\leq 7$, then $E$ is a halfspace. 2. (ii)

The Simons cone \bigl{\{}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}<x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\bigr{\}} is a local minimizer of perimeter in $\mathbb{R}^{8}$ which is not a halfspace.

Also, we recall that these results hold in one dimension higher if we restrict to minimal graphs:

1. (i’)

If E=\bigl{\{}x_{d}\geq h(x_{1},\dots,x_{d-1})\bigr{\}} is a epigraph in $\mathbb{R}^{d}$, $\partial E$ is a minimal surface, and $d\leq 8$ , then $h$ is affine (equivalently, $E$ is a halfspace). 2. (ii’)

There is a non-affine entire minimal graph in dimension $d=9$.

These assertions combine several classical results. The main contributions leading to (i)-(ii)-(i’)-(ii’) are the landmark papers of De Giorgi [12, 13] (improvement of flateness – Bernstein theorem for minimal graphs), Simons [31] (classification of stable minimal cones), and Bombieri, De Giorgi, and Giusti [5] (existence of a nontrivial minimal graph in dimension $d=9$, and minimizing property of the Simons cone).

Note that, in the assumptions of Conjecture 1.1, the function $u$ satisfies $\partial_{x_{d}}u>0$, a condition that implies that the super-level sets $\{u\geq\lambda\}$ are epigraphs. Thus, if we assume that $d\leq 8$, it follows by (ii’) and the discussion in Section 1.2 that the level sets of $u_{\varepsilon}(x)=u(\varepsilon^{-1}x)$ should be close to a hyperplane for $\varepsilon\ll 1$. Since

[TABLE]

this means that all blow-downs of $\{u=\lambda\}$ (i.e., all possible limit points of $\varepsilon\{u=\lambda\}$ as $\varepsilon\downarrow 0$) are hyperplanes. Hence, the conjecture of De Giorgi asserts that, for this to be true, the level sets of $u$ had to be already hyperplanes.

1.4. Results on the De Giorgi conjecture

Conjecture 1.1 was first proved, about twenty years after it was raised, in dimensions $d=2$ and $d=3$, by Ghoussoub and Gui [19] and Ambrosio and Cabré [3], respectively. Almost ten years later, in the celebrated paper [25], Savin attacked the conjecture in the dimensions $4\leq d\leq 8$, and he succeeded in proving it under the additional assumption

[TABLE]

Short after, Del Pino, Kowalczyk, and Wei [15] established the existence of a counterexample in dimensions $d\geq 9$ .

It is worth mentioning that the extra assumption (1.3) in [25] is only used to guarantee that $u$ is a local minimizer of $\mathcal{E}_{1}$. Indeed, while in the case of minimal surfaces epigraphs are automatically minimizers of the perimeter, the same holds for monotone solutions of (1.1) only under the additional assumption (1.3).

1.5. Monotone vs. stable solutions

Before introducing the problem investigated in this paper, we make a connection between monotone and stable solutions.

It is well-known (see [2, Corollary 4.3]) that monotone solutions to (1.1) in $\mathbb{R}^{d}$ are stable solutions, i.e., the second variation of $\mathcal{E}_{1}$ is nonnegative. Actually, in the context of monotone solutions it is natural to consider the two limits

[TABLE]

which are functions of the first $d-1$ variables $x_{1},\dots,x_{d-1}$ only, and one can easily prove that $u^{\pm}$ are stable solutions of (1.1) in $\mathbb{R}^{d-1}$. If one could show that these functions are 1D, then the results of Savin [25] would imply that $u$ was also 1D.

In other words, the following implication holds:

[TABLE]

1.6. Boundary reaction and line tension effects

A natural variant of the Ginzburg-Landau energy, first introduced by Alberti, Bouchitté, and Seppecher in [1] and then studied by Cabré and Solà-Morales in [8], consists in studying a Dirichlet energy with boundary potential on a half space $\mathbb{R}^{d+1}_{+}:=\{x_{d+1}>0\}$ (the choice of considering $d+1$ dimensions will be clear by the discussion in the next sections). In other words, one considers the energy functional

[TABLE]

where $F:\mathbb{R}\to\mathbb{R}$ is some potential. Then, the Euler-Lagrange equation corresponding to $\mathcal{J}$ is given by

[TABLE]

where $f=F^{\prime}$, and $\partial_{\nu}U=-\partial_{x_{d+1}}U$ is the exterior normal derivative. When $f(U)=\sin(c\,U)$, $c\in\mathbb{R}$, the above problem is called the Peierls-Navarro equation and appears in a model of dislocation of crystals [20, 32]. Also, the same equation is central for the analysis of boundary vortices for soft thin films in [21].

1.7. Non-local interactions

To state the analogue of the De Giorgi conjecture in this context we first recall that, for a harmonic function $V$, the energy $\mathcal{J}$ can be rewritten in terms of its trace $v:=V|_{x_{d+1}=0}$. More precisely, a classical computation shows that (up to a multiplicative dimensional constant) the Dirichlet energy of $V$ is equal to the $H^{1/2}$ energy of $v$:

[TABLE]

(see for instance [10]). Hence, instead of $\mathcal{J}$, one can consider the energy functional

[TABLE]

and because harmonic functions minimize the Dirichlet energy, one can easily prove that

[TABLE]

Hence, in terms of the function $u$, the Euler-Lagrange equation (1.4) corresponds to the first variation of $\mathcal{E}$, namely

[TABLE]

where

[TABLE]

1.8. $\Gamma$-convergence of nonlocal energies to the classical perimeter, and the De Giorgi conjecture for the $1/2$-Laplacian

Analogously to what happens with the classical Allen-Cahn equation, there is a connection between solutions of $(-\Delta)^{1/2}u=u-u^{3}$ and minimal surfaces. Namely, if $u$ is a local minimizer of $\mathcal{E}$ in $\mathbb{R}^{d}$ with $F(u)=\frac{1}{4}(1-u^{2})^{2}$, then the rescaled functions $u_{\varepsilon}(x)=u(\varepsilon^{-1}x)$ are local minimizers of the $\varepsilon$-energy

[TABLE]

As happened for the energies $\mathcal{E}_{1,\varepsilon}$ in (1.2), the papers [1, 22] established the $\Gamma$-convergence of $\mathcal{E}_{\varepsilon}$ to the perimeter functional as $\varepsilon\downarrow 0$, as well as the existence of a subsequence $u_{\varepsilon_{k}}$ such that

[TABLE]

where $E$ is a local minimizer of the perimeter in $\mathbb{R}^{d}$. Moreover, Savin and Valdinoci [28] proved density estimates for minimizers of $\mathcal{E}_{\varepsilon}$, implying that $\{u_{\varepsilon_{k}}\geq\lambda\}$ converge locally uniformly to $E$ for each fixed $\lambda\in(-1,1)$.

Hence, the discussion in Section 1.3 motivates the validity of the De Giorgi conjecture when $-\Delta$ is replaced with $(-\Delta)^{1/2}$, namely:

Conjecture 1.2**.**

Let $u\in C^{2}(\mathbb{R}^{d})$ be a solution of the fractional Allen-Cahn equation

[TABLE]

satisfying $\partial_{x_{d}}u>0$. Then, if $d\leq 8$, all level sets $\{u=\lambda\}$ of $u$ must be hyperplanes.

In this direction, Cabré and Solà-Morales proved the conjecture for $d=2$ [8]. Later, Cabré and Cinti [6] established Conjecture 1.2 for $d=3$. Very recently, under the additional assumption (1.3), Savin has announced in [26] a proof of Conjecture 1.2 in the remaining dimensions $4\leq d\leq 8$. Thanks to the latter result, the relation between monotone and stable solutions explained in Section 1.5 holds also in this setting.

1.9. Stable solutions vs. stable minimal surfaces

Exactly as in the setting of Conjecture 1.1, given $u$ as in Conjecture 1.2 it is natural to introduce the two limit functions $u^{\pm}:=\lim_{x_{d}\to\pm\infty}u$. These functions depend only on the first $d-1$ variables $x_{1},\dots,x_{d-1}$, and are stable solutions of (1.6) in $\mathbb{R}^{d-1}$.

As mentioned at the end of last section, the classification of stable solutions to (1.6) in $\mathbb{R}^{d-1}$, $3\leq d-1\leq 7$, together with the improvement of flatness for $(-\Delta)^{1/2}u=u-u^{3}$ announced in [26], would imply the full Conjecture 1.2 in $\mathbb{R}^{d}$.

The difficult problem of classifying stable solutions of (1.6) (or of (1.1)) is connected to the following well-known conjecture for minimal surfaces:

Conjecture 1.3**.**

Stable embedded minimal hypersurfaces in $\mathbb{R}^{d}$ are hyperplanes as long as $d\leq 7$.

A positive answer to this conjecture is only known to be true in dimension $d=3$, a result of Fischer-Colbrie and Schoen [18] and Do Carmo and Peng [16]. Note that, for minimal cones, the conjecture is true (and the dimension 7 sharp) by the results of Simons [31] and Bombieri, De Giorgi, and Giusti [5].

Conjecture 1.3 above suggests a “stable De Giorgi conjecture”:

Conjecture 1.4**.**

Let $u\in C^{2}(\mathbb{R}^{d})$ be a stable solution of (1.1) or of (1.6). Then, if $d\leq 7$, all level sets $\{u=\lambda\}$ of $u$ must be hyperplanes.

As explained before, the validity of this conjecture would imply both Conjectures 1.1 and 1.2.

1.10. Results of the paper

As of now, Conjecture 1.4 has been proved only for $d=2$ (see [4, 19] for (1.1), and [8] for (1.6)). The main result of this paper establishes its validity for (1.6) and $d=3$, a case that heuristically corresponds to the classification in $\mathbb{R}^{3}$ of stable minimal surfaces of [18]. Note that, for the classical case (1.1), Conjecture 1.4 in the case $d=3$ is still open.

This is our main result:

Theorem 1.5**.**

Let $u$ be a stable solution of (1.5) with $d=3$ such that $|u|\leq 1$, and assume that $f\in C^{0,\alpha}([-1,1])$ for some $\alpha>0$. Then $u$ is 1D profile, namely, $u(x)=\phi(e\cdot x)$ for some $e\in\mathbb{S}^{2}$, where $\phi:\mathbb{R}\to\mathbb{R}$ is a nondecreasing bounded stable solution to (1.5) in dimension one.

As explained before, as an application of Theorem 1.5 and the improvement of flatness for $(-\Delta)^{1/2}u=u-u^{3}$ announced in [26], we obtain the following:

Corollary 1.6**.**

Conjecture 1.2 holds true in dimension $d=4$.

A key ingredient behind the proof of Theorem 1.5 is the following general energy estimate which holds in every dimension $d\geq 2$:

Proposition 1.7**.**

Let $R\geq 1$, $M_{o}\geq 2$, and $\alpha\in(0,1)$. Assume that $u$ be a stable solution of

[TABLE]

where $f:[-1,1]\to\mathbb{R}$ satisfies $\|f\|_{C^{0,\alpha}([-1,1])}\leq M_{o}$. Then there exists a constant $C>0$, depending only on $d$ and $\alpha$, such that

[TABLE]

and

[TABLE]

Note that the estimates in Proposition 1.7 differ from being sharp by a factor $\log(M_{o}R)$ (just think of the case when $u$ is a 1D profile). However, for stable solutions of (1.7) in $\mathbb{R}^{3}$ we are able to bootstrap these non-sharp estimates to sharp ones, from which Theorem 1.5 follows easily.

1.11. Structure of the paper

In the next section we collect all the basic estimates needed for the proof of Proposition 1.7. Then, in Section 3 we prove Proposition 1.7. Finally, in Section 4 we prove Theorem 1.5.

Acknowledgments: both authors are supported by ERC Grant “Regularity and Stability in Partial Differential Equations (RSPDE)”.

2. Ingredients of the proofs

We begin by introducing some notation.

Given $R>0$, we define the energy of a function inside $B_{R}\subset\mathbb{R}^{d}$ as

[TABLE]

where $B_{R}^{c}=\mathbb{R}^{d}\setminus B_{R}$, and $F$ is a primitive of $f$. Note that equation (1.7) is the condition of vanishing first variation for the energy functional $\mathcal{E}(\,\cdot\,;B_{R})$.

We say that a solution $u$ of (1.7) is stable if the second variation at $u$ of $\mathcal{E}$ is nonnegative, that is

[TABLE]

Also, we say that $u$ is stable in $\mathbb{R}^{d}$ if it is stable in $B_{R}$ for all $R\geq 1$.

An important ingredient in our proof consists in considering variations of a stable solution $u$ via a suitable smooth $1$-parameter family of “translation like” deformations. This kind of idea has been first used by Savin and Valdinoci in [27, 29], and then in [11, 7]. More precisely, given $R\geq 3$, consider the cut-off functions

[TABLE]

[TABLE]

[TABLE]

where $R^{*}:=\exp(\sqrt{\log R})$.

For a fixed unit vector $\boldsymbol{v}\in\mathbb{S}^{n-1}$ define

[TABLE]

Then, given a function $v:\mathbb{R}^{d}\to\mathbb{R}$ and $t\in(-1,1)$ with $|t|$ small enough (so that $\Psi_{t,\boldsymbol{v}}^{i}$ is invertible), we define the operator

[TABLE]

Also, we use $\mathcal{E}^{\rm Sob}$ and $\mathcal{E}^{\rm Pot}$ to denote respectively the fractional Sobolev term and the Potential term appearing in the definition of $\mathcal{E}$:

[TABLE]

[TABLE]

We shall use the following bounds:

Lemma 2.1**.**

There exists a dimensional constant $C$ such that the following hold for all $v:\mathbb{R}^{d}\to\mathbb{R}$, $|t|$ small, and $\boldsymbol{v}\in\mathbb{S}^{d-1}$:

(1) We have

[TABLE]

(2) For $R=2^{2k}$, $k\geq 1$, we have

[TABLE]

(3) For $R\geq 4$,

[TABLE]

Proof.

The lemma follows as in [11, Lemma 2.1] and [7, Lemma 2.3]. However, since we do not have a precise reference for the estimates that we need, we give a sketch of proof. Note that, by approximation, it suffices to consider the case when $v\in C^{2}_{c}(\mathbb{R}^{d})$.

First observe that, since $\boldsymbol{v}$ has unit norm, the Jacobian of the change of variables $z\mapsto\Psi_{t,\boldsymbol{v}}^{i}(z)$ is given by

[TABLE]

Set

[TABLE]

Then, performing the change of variables $x:=\Psi_{t,\boldsymbol{v}}^{i}(z)$, we get

[TABLE]

thus

[TABLE]

Hence, we only need to estimate the second order incremental quotient of $\mathcal{E}^{\rm Sob}$. To this aim, using the same change of variable and setting

[TABLE]

and $K(z):=|z|^{-(d+1)}$, we have (note that $\Psi_{t,\boldsymbol{v}}^{i}$ preserves $B_{R^{i}}$)

[TABLE]

Recalling that \Psi^{i}_{t,\boldsymbol{v}}(y)-\Psi^{i}_{t,\boldsymbol{v}}(\overline{y})=y-\overline{y}+t\big{(}\varphi^{i}(y)-\varphi^{i}(\overline{y})\big{)}\boldsymbol{v} and defining

[TABLE]

as in the proof of [11, Lemma 2.1] we have, for $|t|$ small,

[TABLE]

and

[TABLE]

Then, using (2.9), (2.7), (2.10), and (2.11), and decomposing $A_{R}=A_{2^{2k}}=A_{2^{k}}\cup\left(\cup_{j=1}^{k}A_{2^{k+j}}\setminus A_{2^{k+j-1}}\right)$ when $i=1,$ an easy computation yields

[TABLE]

(see the proof of [11, Lemma 2.1] for more details). Therefore (2.4) and (2.5) follow.

The proof of (2.6) needs a more careful estimate. For $\rho>0$, let us denote

[TABLE]

Note that

[TABLE]

Observing that in the complement of $A_{\rho}$ we have $|y|\geq\rho$ and $|\overline{y}|\geq\rho$ and using (2.9), (2.7), (2.10), and (2.11) we obtain

[TABLE]

where

[TABLE]

so (2.6) follows. ∎

The following is a basic BV estimate in $B_{1/2}$ for stable solutions in a ball.

Lemma 2.2**.**

Let $i\in\{0,1,2\}$, $R^{i}$ as in (2.8), and let $u\in C^{1,\alpha}(\overline{B_{R^{i}}})$ be a stable solution to $(-\Delta)^{1/2}u+f(u)=0$ in $B_{R^{i}}$ with $|u|\leq 1$. Assume there exists $\eta>0$ such that, for $|t|$ small enough, we have

[TABLE]

Then

[TABLE]

and

[TABLE]

for some dimensional constant $C$.

Proof.

The proof is similar to the ones of [11, Lemmas 2.4 and 2.5] or [7, Lemma 2.5 and 2.6]. The key point is to note that, since $u$ is stable,

[TABLE]

hence (2.12) implies

[TABLE]

for $|t|$ small enough.

On the other hand, still by stability, the two functions

[TABLE]

satisfy

[TABLE]

Hence, combining these inequalities with the identity

[TABLE]

we obtain

[TABLE]

Noticing that $\mathcal{P}^{i}_{t,\boldsymbol{v}}u(x)=u(x-t\boldsymbol{v})$ for $x\in B_{1/2}$ and that $|x-y|^{-d-1}\geq 1$ for $x,y\in B_{1/2}$ we obtain the bound

[TABLE]

for all $|t|$ small enough, so (2.13) follows by letting $t\to 0$.

In other words, if we define

[TABLE]

we have proved that $\min\{A_{\boldsymbol{v}}^{+},A_{\boldsymbol{v}}^{-}\}^{2}\leq A_{\boldsymbol{v}}^{+}A_{\boldsymbol{v}}^{-}\leq 2\eta$. In addition, since $|u|\leq 1$, by the divergence theorem

[TABLE]

Combining these bounds, this proves that

[TABLE]

from which (2.14) follows immediately. ∎

We now recall the following general lemma due to Simon [30] (see also [11, Lemma 3.1]):

Lemma 2.3**.**

Let $\beta\in\mathbb{R}$ and $C_{0}>0$. Let $\mathcal{S}:\mathcal{B}\rightarrow[0,+\infty]$ be a nonnegative function defined on the class $\mathcal{B}$ of open balls $B\subset\mathbb{R}^{n}$ and satisfying the following subadditivity property:

[TABLE]

Also, assume that $\mathcal{S}(B_{1})<\infty.$ Then there exists $\delta=\delta(n,\beta)>0$ such that if

[TABLE]

then

[TABLE]

where $C$ depends only on $d$ and $\beta$.

Finally, we state an optimal bound on the $H^{1/2}$ norm of the mollification of a bounded function with the standard heat kernel, in terms of the $BV$ norm and the parameter of mollification (see [17, Lemma 2.1] for a proof):

Lemma 2.4**.**

Let $H_{d,t}(x):=(4\pi t)^{-d/2}e^{-|x|^{2}/4t}$ denote the heat kernel in $\mathbb{R}^{d}$. Given $u\in BV(\mathbb{R}^{d})$ with $|u|\leq 1$, set $u_{\varepsilon}:=u\ast H_{d,\varepsilon^{2}}$. Then, for $\varepsilon\in(0,1/2)$, we have

[TABLE]

where $C$ is a dimensional constant.

3. Proof of Proposition 1.7

As a preliminary result we need the following (sharp) interpolation estimate.

Lemma 3.1**.**

Let $u:\mathbb{R}^{d}\to\mathbb{R}$ be a bounded function, with $|u|\leq 1$. Assume that $u$ is Lipschitz in $B_{2}$, with $\|\nabla u\|_{L^{\infty}(B_{2})}\leq L_{o}$ for some $L_{o}\geq 2$. Then

[TABLE]

where $C$ depends only on $d$.

Proof.

Let $\eta\in C^{\infty}_{c}(B_{2})$, $0\leq\eta\leq 1$, be a radial cutoff function such that $\eta=1$ in $B_{3/2}$ and $\|\nabla\eta\|_{L^{\infty}(\mathbb{R}^{d})}\leq 3$, and set $\widetilde{u}:=\eta u$. Observe that, since $|u|\leq 1$, $0\leq\eta\leq 1$, $\|\nabla u\|_{L^{\infty}(B_{2})}\leq L_{o}$, and $\eta$ is supported inside $B_{2}$, we have (recall that $L_{0}\geq 2$)

[TABLE]

Now, since $|u|\leq 1$, we have

[TABLE]

where $C$ depends only on $d$. On the other hand, it follows by Lemma 2.4 that

[TABLE]

We also observe that, because of (3.1),222The first inequality in (3.4) can be proven using Fourier transform, noticing that

$\widehat{(\widetilde{u}_{\epsilon}-u)}(\xi)=(e^{-\varepsilon^{2}|\xi|^{2}}-1)\hat{u}(\xi),$

and that $\frac{|e^{-\varepsilon^{2}|\xi|^{2}}-1|^{2}}{\varepsilon|\xi|}$ is universally bounded. Indeed,

[\widetilde{u}_{\varepsilon}-\widetilde{u}]_{H^{1/2}(\mathbb{R}^{d})}^{2}=\int|\xi|\,\bigl{|}\widehat{(\widetilde{u}_{\epsilon}-u)}(\xi)\bigr{|}^{2}\,d\xi=\int|\xi|\,\bigl{|}e^{-\varepsilon^{2}|\xi|^{2}}-1\bigr{|}^{2}|\hat{u}(\xi)|^{2}\,d\xi\\ \leq C\varepsilon\int|\xi|^{2}|\hat{u}(\xi)|^{2}\,d\xi=C\varepsilon\|\nabla\widetilde{u}\|_{L^{2}(\mathbb{R}^{d})}^{2}.

[TABLE]

Therefore, choosing $\varepsilon=(L_{o})^{-1}$ in (3.3) and (3.4), and using a triangle inequality, we get (recall that $L_{0}\geq 2$)

[TABLE]

Finally, we note that

[TABLE]

and that (cp. (3.1))

[TABLE]

Hence, recalling (3.2), we obtain

[TABLE]

and the lemma follows. ∎

We can now prove Proposition 1.7.

Proof of Proposition 1.7.

This proof is similar to the proof of Theorem 2.1 in [7] (see also the proof Theorem 1.7 in [11]). Here we need to use, as a new ingredient, the estimate from Lemma 3.1. Throughout the proof, $C$ denotes a generic dimensional constant.

• Step 1. Let $v$ be a solution a stable solution $(-\Delta)^{1/2}v+g(v)=0$ in $B_{3}$ satisfying $|v|\leq 1$ in all of $\mathbb{R}^{d}$.

First, using (2.4) in Lemma 2.1 and then Lemma 2.2 with $i=0$ and $R^{i}=1$, we obtain

[TABLE]

Note that this estimate is valid for every stable solution $v$, independently of the nonlinearity $g$.

On the other hand, note that if $\|g\|_{C^{0,\alpha}([-1,1])}\leq M_{o}$ for some $M_{o}\geq 2$, then by the interior regularity estimates for $(-\Delta)^{1/2}$ we have

[TABLE]

Therefore, combining (3.5) with Lemma 3.1, we obtain

[TABLE]

where we used the inequality $2\sqrt{ab}\leq\delta a+b/\delta$ for $a,b\in\mathbb{R}_{+}$.

• Step 2. For $v$ as in Step 1 and $B_{\rho}(z)\subset B_{1}$ we note that the function

[TABLE]

satisfies $(-\Delta)^{1/2}\widetilde{v}+\widetilde{g}(\widetilde{v})=0$ with $\widetilde{g}(s):=2\rho g(s)$. In particular $\|\widetilde{g}\|_{C^{0,\alpha}([-1,1])}\leq 2\rho M_{o}\leq 2M_{o}$, so estimate (3.7) applied to $\widetilde{v}$ yields

[TABLE]

or equivalently

[TABLE]

Hence taking $\delta$ small enough and using Lemma 2.3 with $\mathcal{S}(B):=\int_{B}|\nabla v|\,dx$ and $\beta:=1-d$, we obtain

[TABLE]

where $C$ depends only on $d$. Also, it follows by Lemma 3.1 and (3.6) that

[TABLE]

• Step 3. If $u$ is a stable solution $u$ of $(-\Delta)^{1/2}u=f(u)$ in $B_{R}$, given $x_{o}\in B_{R/2}$ we consider the function v(x):=u\big{(}x_{o}+\frac{R}{6}x\big{)}. Note that this function satisfies (3.9) and (3.10) with $M_{0}$ replaced by $M_{0}R$, hence the desired estimates follow easily by scaling and a covering argument. ∎

4. Proof of Theorem 1.5

We are given $u$ a stable solution of $(-\Delta)^{1/2}u+f(u)=0$ in $\mathbb{R}^{3}$ with $|u|\leq 1$ and $f\in C^{0,\alpha}$, and we want to show that $u$ is 1D. We split the proof in three steps.

• Step 1. By Proposition 1.7 we have

[TABLE]

for all $R\geq 2$, where $C$ depends only on $f$. Take $k\geq 1$, $R=2^{2(k+1)}$, and $v(x):=u(Rx)$. Note that, by elliptic regularity, $\|\nabla u\|_{L^{\infty}(\mathbb{R}^{3})}\leq C$ for some constant $C$ depending only on $f$, thus $\|\nabla v\|_{L^{\infty}(\mathbb{R}^{3})}\leq CR$. Also, $v$ is still a stable solution of a semilinear equation in all of $\mathbb{R}^{3}$. Hence, using (2.5) in Lemma 2.1 and then Lemma 2.2 with $i=1$ and $R^{i}=R$, we obtain

[TABLE]

On the other hand, using Lemma 3.1 and the bound $\|\nabla v\|_{L^{\infty}(\mathbb{R}^{3})}\leq CR$, we have

[TABLE]

Thus, recalling that $R=2^{2(k+1)}=4\cdot 2^{2k}$, $v(x)=u(Rx)$, rewriting (4.2) and (4.3) in terms of $u$ we get (here we use that $d=3$)

[TABLE]

• Step 2. Given $j\geq 1$ set

[TABLE]

so that (4.4) can be rewritten as

[TABLE]

where $C_{f}$ depends only on $f$. We claim that

[TABLE]

for some constant $M$ depending only on $f$.

Indeed, assume by contradiction that $\mathcal{A}(k_{0})\geq M$ for some large constant $M$ to be chosen later. Rewriting (4.5) as

[TABLE]

then, provided $M\geq\frac{1}{c_{f}}$, if $\mathcal{A}(k_{0})\geq M$ we find

[TABLE]

This implies that there exists $k_{1}\in\{k_{0}+1,\ldots,3k_{0}\}$ such that

[TABLE]

that is

[TABLE]

Hence, choosing $M$ large enough so that $\widetilde{M}:=(c_{f}M)^{2}\geq\frac{1}{c_{f}}$, we can repeat exactly the same argument as above with $M$ replaced by $\widetilde{M}$ and $k_{0}$ replaced by $k_{1}$ in order to find $k_{2}\in\{k_{1}+1,3k_{1}\}$ such that

[TABLE]

Iterating further we find $k_{1}<k_{2},<k_{3}<\ldots<k_{m}<\ldots$ such that $k_{m+1}\leq 3k_{m}$ and

[TABLE]

Now, ensuring that $M$ is large enough so that $\theta:=c_{f}^{2}M>1$, we obtain

[TABLE]

On the other hand, recalling (4.1) and using that $k_{m}\leq 3^{m}k_{0}$, we have

[TABLE]

The linear bound from (4.8) clearly contradicts the exponential growth in (4.7) for $m$ large enough. Hence, this provides the desired contradiction and proves (4.6)

• Step 3. Rephrasing (4.6), we proved that

[TABLE]

for all $R\geq 2$, where $C$ depends only on $f$. In other words, we have obtained an optimal energy estimate in large balls $B_{R}$ (note that 1D profiles saturates (4.9)). Having improved the energy bound of Proposition 1.7 from $R^{2}\log^{2}R$ to (4.9), we now conclude that $u$ is a 1D profile as follows.

Given $\boldsymbol{v}\in\mathbb{S}^{d-1}$ and using the perturbation $\mathcal{P}^{2}_{t,\boldsymbol{v}}$ as in (2.2)-(2.3), it follows by (2.6), (2.12), and (2.13) that

[TABLE]

Hence, taking the limit as $R\to\infty$ we find that

[TABLE]

thus

[TABLE]

Since this argument can be repeated changing the center of the ball $B_{1/2}$ with any other point, by a continuity argument we obtain that

[TABLE]

Thanks to this fact, we easily conclude that $u$ is a 1D monotone function, as desired. ∎