Gradient estimates for the Allen-Cahn equation on Riemannian manifolds

TL;DR

This paper derives gradient estimates for positive solutions to the Allen-Cahn equation on Riemannian manifolds and applies these results to establish a Liouville theorem under nonnegative Ricci curvature.

Contribution

It provides new gradient estimates for the Allen-Cahn equation on Riemannian manifolds and proves a Liouville theorem for nonnegative Ricci curvature cases.

Findings

Gradient estimates for solutions on manifolds

Liouville theorem for nonnegative Ricci curvature

Extension of classical results to Riemannian setting

Abstract

In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.

Equations

$$\Delta u+(1-u^2)u=0,$$

$$\frac{|\nabla u{|}^2}{u^2}+\frac{2}{3}(1-u^2)$$

$$\le \frac{n}{1-\epsilon }\left(\frac{2C_1^2+(n-1)C_1^2(1+R\sqrt{K(2R)})+C_2}{R^2}+\frac{2n}{(1-\epsilon )}\frac{C_1^2}{R^2}+2K(2R)\right)$$

$$\frac{|\nabla u{|}^2}{u^2}+s(1-u^2)$$

$$\le \frac{n{s}^2}{2(1-\epsilon )}\left(\frac{n}{4\left(1-\epsilon \right)}\frac{s^2}{\left(sq+s-1\right)}\frac{C_1^2}{R^2}\frac{2C_1^2+(n-1)C_1^2(1+R\sqrt{K(2R)})+C_2}{R^2}\right)$$

$$+\frac{n{s}^2}{2(1-\epsilon )(s-1)}K(2R)+\frac{s}{q}\sqrt{\frac{n}{2(1-\epsilon )}}{C}^2$$

$$\frac{|\nabla u{|}^2}{u^2}+2(1-u^2)$$

$$+4nK(2R)+54n\sqrt{n}{C}^2.$$

$$\frac{|\nabla u{|}^2}{u^2}+\frac{2}{3}(1-u^2)\le \frac{2nk}{1-\epsilon }.$$

$$\frac{|\nabla u{|}^2}{u^2}+\frac{2}{3}(1-u^2)\le 2nk.$$

$$|\nabla u{|}^2\le 2nk.$$

$$\frac{|\nabla u{|}^2}{u^2}+s(1-u^2)\le \frac{n{s}^{2}k}{2(1-\epsilon )(s-1)}+\frac{s}{q}\sqrt{\frac{n}{2(1-\epsilon )}}{C}^2.$$

$$\frac{|\nabla u{|}^2}{u^2}\le 4nk+(54n\sqrt{n}+2)C^2.$$

$$|\nabla u{|}^2\le (4nk+(54n\sqrt{n}+2)C^2)C^2.$$

$$\Delta u=f(u),$$

$$\Delta u+u^p-u^q=0,$$

$$W(x)=u^{-q},$$

$$\nabla W=-q{u}^{-q-1}\nabla u,$$

$$|\nabla W{|}^2=q^2{u}^{-2q-2}|\nabla u{|}^2,$$

$$\frac{|\nabla W{|}^2}{W^2}=q^2{u}^{-2}|\nabla u{|}^2,$$

$$\Delta W$$

$$=\frac{q+1}{q}\frac{|\nabla W{|}^2}{W}+qW-q{W}^{\frac{q-2}{q}}.$$

$$F(x)=\frac{|\nabla W{|}^2}{W^2}+\alpha (1-W^{-2/q}),$$

$$\nabla F(x)=\frac{\nabla |\nabla W{|}^2}{W^2}-\frac{2|\nabla W{|}^2\nabla W}{W^3}+\frac{2\alpha }{q}{W}^{-(q+2)/q}\nabla W,$$

$$\Delta F(x)$$

$$+6\frac{|\nabla W{|}^4}{W^4}-2\frac{|\nabla W{|}^2\Delta W}{W^3}-\frac{2\alpha (q+2)}{q^2}\frac{W^{-2/q}}{W^2}|\nabla W{|}^2$$

$$+\frac{2\alpha }{q}{W}^{-(q+2)/q}\Delta W.$$

$$\frac{2⟨\nabla W,\Delta \nabla W⟩}{W^2}$$

$$=\frac{4(q+1)}{q}\frac{⟨{\nabla }^{2}W,\nabla W\otimes \nabla W⟩}{W^3}-\frac{2(q+1)}{q}\frac{|\nabla W{|}^4}{W^4}$$

$$+\frac{2|\nabla W{|}^2}{W^2}\left[q-(q-2)W^{-2/q}\right]+\frac{2\text{Ric}⟨\nabla W,\nabla W⟩}{W^2},$$

$$-\frac{2|\nabla W{|}^2\Delta W}{W^3}=-\frac{2(q+1)}{q}\frac{|\nabla W{|}^4}{W^4}-2q\frac{|\nabla W{|}^2}{W^2}+2q{W}^{-\frac{2}{q}}\frac{|\nabla W{|}^2}{W^2},$$

$$\frac{2\alpha }{q}{W}^{-(q+2)/q}\Delta W=\frac{2\alpha (q+1)}{q^2}{W}^{-2/q}\frac{|\nabla W{|}^2}{W^2}+2\alpha W^{-2/q}-2\alpha W^{-4/q}.$$

$$\frac{2\epsilon |{\nabla }^{2}W{|}^2}{W^2}+\frac{2}{\epsilon }\cdot \frac{|\nabla W{|}^4}{W^4}\ge 4\frac{⟨{\nabla }^{2}W,\nabla W\otimes \nabla W⟩}{W^3}.$$

$$\frac{2|{\nabla }^{2}W{|}^2}{W^2}-8\frac{⟨{\nabla }^{2}W,\nabla W\otimes \nabla W⟩}{W^3}+6\frac{|\nabla W{|}^4}{W^4}$$

$$+\left(6-\frac{2}{\epsilon }\right)\frac{|\nabla W{|}^4}{W^4},$$

Full text

1. Introduction

The Allen-Cahn equation

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has its origin in the gradient theory of phase transitions [1], and has attracted a lot of attentions in the last decades. The famous De Giorgi conjecture states that for $n\leq 8$, any entire solution to (1.1) in $\mathbb{R}^{n}$ with $|u|<1$ which is monotone in one direction should be one-dimensional [6]. The conjecture was proved in dimension 2 by Ghoussoub-Gui [8] and in dimension 3 by Ambrosio-Cabré [2], and in dimensions $4\leq n\leq 8$ by Savin [16], under an extra assumption. For $n\geq 9$, the conjecture is false [7].

Solutions to the Allen-Cahn equation have the intricate connection to the minimal surface theory. There are many results in the literature, such as, solutions concentrating along non-degenerate, minimal hypersurfaces of a compact manifold were found in [14]. So the equation is also an interesting topic for geometry.

The gradient estimate is a useful method in the study of elliptic and parabolic equations. It was originated by Yau [20], Cheng-Yau[5], and Li-Yau [11], and was extended by many authors, say Li[9], Negrin[13], Souplet-Zhang [17], Ma [10], Yang [18, 19], Cao [4] for various purposes. In this paper, we consider bounded positive solutions to Eq.(1.1) and get the following theorem.

Theorem 1.1**.**

Let $M$ be a complete noncompact $n$-dimensional Riemannian manifold without boundary. Denote by $B_{p}(2R)$ the geodesic ball of radius $2R$ around $P\in M$. Suppose $\textup{Ric}\geq-K(2R)$ in $B_{p}(2R)$ with $K(2R)\geq 0$, $u$ is a bounded positive smooth solution of (1.1) on $M$ $u\leq C$ where $C$ is a positive constant.

(1) If $C\leq 1$, then we have

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on $B_{p}(R)$, where $C_{1}$, $C_{2}$ are positive constants, $0<\varepsilon<1$.

(2) If $C>1$, then we have

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on $B_{p}(R)$, where $C_{1}$, $C_{2}$ are positive constants; $0<\varepsilon<1$, $s>1$, $q>0$ such that $\frac{2(1-\varepsilon)}{n}\frac{s-1}{sq}\geq\frac{1}{\varepsilon}-1+\frac{(3s-1)^{2}}{2}$. In particular, we can choose $q=\frac{2(1-\varepsilon)(s-1)}{ns\left[\frac{1}{\varepsilon}-1+\frac{(3s-1)^{2}}{2}\right]}$. Taking $s=2$ and $\varepsilon=1/2$, we get

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As a consequence of Theorem 1.1, we have the following:

Corollary 1.1**.**

Let $M$ be a complete noncompact $n$-dimensional Riemannian manifold with Ricci tensor $\textup{Ric}\geq-{k}\,({k\geq 0})$. Suppose $u$ is a positive solution of (1.1) and $u\leq C$.

(1) If $C\leq 1$, we have

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Letting $\varepsilon$ approach zero, we get

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Furthermore,

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(2) If $C>1$, we have

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In particular, choosing $s=2$ and $\varepsilon=1/2$, we have

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Furthermore,

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For an application of Corollary 1.1, we get the following Liouville type theorem:

Theorem 1.2**.**

Let $M$ be a complete noncompact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature. If $u$ is a solution of (1.1) with $0<u\leq 1$, then $u$ is equal to $1$ identically on $M$.

In general, let $F\in C^{2}(\mathbb{R})$ be a nonnegative function and $u\in C^{3}({\mathbb{R}^{n}})$ a bounded entire solution in $\mathbb{R}^{n}$ of the equation

[TABLE]

where $f=F^{{}^{\prime}}$ is the first derivative of $F$. L. Modica [12] proved that $|\nabla u|^{2}(x)\leq 2F(u(x))$ for every $x\in\mathbb{R}^{n}$. Later Ratto-Rigoli [15] extended Modica’s result to manifolds with nonnegative Ricci curvature. Also the conclusion of Theorem 1.2 can be deduced from the result of Ratto and Rigoli by setting $F(u)=\frac{1}{4}u^{4}-\frac{1}{2}u^{2}+\frac{1}{4}$. However our result gives an explicit bound of $|\nabla u|$ in the case $k\neq 0$. In addition, Corollary 1.1 implies that the equation (1.1) does not admit an entire solution with values in $(0,1)$ on manifolds with nonnegative Ricci curvature. The method in this paper can be also applied to the equation

[TABLE]

where $p,q\in\mathbb{R}$.

The rest of the paper is arranged as follows. In Section 2, we prove a basic lemma. In Section 3, we prove the main results.

2. Basic Lemma

We consider

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as the one defined in [9], where $q$ is a positive constant to be chosen later. A straightforward computation shows that

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[TABLE]

[TABLE]

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We introduce the function

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where $\alpha$ is a positive constant to be fixed later.

Now we calculate

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Noting (2.2) we have

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[TABLE]

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By the Hölder inequality, we have

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Hence

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where $0<\varepsilon<1.$

Using the fact $|\nabla^{2}W|^{2}\geq\frac{1}{n}{\left(\Delta W\right)^{2}}$, we get

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By (2.4),

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From (2.5) to (2.10), we obtain

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It follows from (2.2) and (2.3) that

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Set $\alpha=sq^{2}$, then

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Substituting (2.13) into (2.11) gives

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We get the following lemma.

Lemma 2.1**.**

Let $M$ be a complete noncompact $n$-dimensional Riemannian manifold without boundary. If $F$ is defined by (2.3) where $\alpha=sq^{2}$, then we have

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3. Proof of Main Results

Proof of Theorem 1.1. Chose a cut-off function $\chi$ $\in C^{2}[0,+\infty)$ such that $\chi(r)=1$ for $r\leq 1$, $\chi(r)=0$ for $r>2$, and $0\leq\chi(r)\leq 1$. In addition, we require $\chi$ satisfies $-C_{1}\leq\chi^{-1/2}(r)\chi^{\prime}(r)\leq 0$ and $\chi^{\prime\prime}(r)\geq-C_{2}$, where $C_{1},C_{2}$ are positive constants.

For a fixed point $p$, denote by $r(x)$ the geodesic distance between $x$ and $P$. Define

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It is clear that

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By the Laplacian comparison theorem, we get

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Now we consider the function $\phi(x)F(x)$. By the argument of Calabi[3], we assume that the function $\phi(x)F(x)$ is smooth in $B_{P}(2R)$. Let $z$ be the point where $\phi F$ achieves its maximum in $B_{P}(2R)$. We can assume that $\lambda:=\phi(z)F(z)>0$ since the theorem is obviously true if $\lambda\leq 0$. Then we have

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and

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at the point $z$,

Using Eq.(3.1), we have

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By (3.2), we have

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Thus we obtain

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at $z$.

Then for

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we have

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Multiplying both sides of (2.14) by $\phi^{2}$, we obtain at $z$,

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We consider two cases: (1) $C\leq 1$ and (2) $C>1$.

(1) Since $u\leq 1$, it is easy to see that

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and

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if $s\geq\frac{2}{3}$.

Substituting (3.4), (3.5) into (3.3), and choosing $s=\frac{2}{3}$ and $q>0$ small enough such that $\frac{(1-\varepsilon)}{n}\frac{(sq+s-1)^{2}}{s^{2}q^{2}}\geq\frac{1}{\varepsilon}-1$, then we have

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We take the similar technique as in [10]. Clearly,

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Combining (3.6) and (3.7), we arrive at

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It follows that

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In other words, we get

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Note that

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Substituting (3.9) into (3.8), we get

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Then we get

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(2) By the condition on Ricci curvature, we derive

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By Hölder’s inequality, we get

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and

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By (3.4),

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Using Hölder’s inequality again gives

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Choose $s>1$ and $q>0$ such that $\frac{2(1-\varepsilon)}{n}\frac{s-1}{sq}\geq\frac{1}{\varepsilon}-1+\frac{(3s-1)^{2}}{2}$. Then (3.3) becomes

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whence

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Thus

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Combining (3.10) and (3.11), we conclude the theorem. $\hfill\Box$

Proof of Corollary 1.1. Passing to the limit $R\rightarrow+\infty$ in the estimates of Theorem 1.1, we get the desired results. $\hfill\Box$

Proof of Theorem 1.2. Suppose that $M$ is a complete noncompact Riemannian manifold with nonnegative Ricci curvature. If $u$ is a solution of (1.1) on $M$ and $0<u\leq 1$, then by Corollary 1.1, we get

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It follows that $|\nabla u|\equiv 0$ and $u\equiv 1$. This concludes Theorem 1.2. $\hfill\Box$