On the Łojasiewicz–Simon gradient inequality on submanifolds

Abstract

We provide sufficient conditions for the Łojasiewicz–Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. Our result provides a tool to study asymptotic properties of quasilinear parabolic equations with (nonlinear) constraints.

Introduction

In real algebraic geometry, the Łojasiewicz inequality is a remarkable result describing the particular behavior of an analytic function near a critical point.

Theorem 1.1 Łojasiewicz inequality, [27, Théorème 4]

Let $U\subset {\mathbb{R}}^n$ be open. If $\mathcal{E}\in {\mathcal{C}}^{\omega }(U;\mathbb{R})$ and $\overline{u}\in U$ satisfies $\mathrm{\nabla }\mathcal{E}(\overline{u})=0$, then there exist $C,\sigma >0$ and $\theta \in (0,\frac{1}{2}]$ such that for all $\Vert u-\overline{u}\Vert \le \sigma$, we have

$$|\mathcal{E}(u)-\mathcal{E}(\overline{u}){|}^{1-\theta }\le C\Vert \mathrm{\nabla }\mathcal{E}(u)\Vert .$$

Throughout this article, we write ${\mathcal{C}}^{\omega }(U;X)$ for the set of real analytic functions from an open set U of a Banach space V into another Banach space X. All vector spaces are understood to be over the field of real numbers $\mathbb{R}$. The space of bounded linear operators between two normed spaces X and Y is denoted by $\mathcal{L}(X,Y)$ and we write $X^{⁎}:=\mathcal{L}(X,\mathbb{R})$ for the continuous dual of X.

In ${\mathbb{R}}^n$, inequality (1) was discovered and proven by S. Łojasiewicz in his famous works on semianalytic and subanalytic sets, [27], [28]. Since then, Theorem 1.1 has been used as a celebrated tool to prove convergence results for the gradient flow of analytic energies on finite dimensional spaces (see [29]). The pioneering work of L. Simon in [33] extended inequality (1) to certain energy functions on infinite-dimensional function spaces using Lyapunov–Schmidt reduction and, in honor of his significant contributions, the inequality is nowadays often called Łojasiewicz–Simon gradient inequality. In more recent work by Kurdyka [22], Łojasiewicz's convergence result has been extended to a larger class of functions via the Kurdyka–Łojasiewicz inequality. Over the last decades, gradient inequalities like (1) have been extensively studied in various situations to analyze the long time behavior of gradient flows, see for instance [12], [13], [15], [18], [32]. In [20], [21], this is also done for second order evolution equations. Loosely speaking, whenever an energy $\mathcal{E}$ satisfies a Łojasiewicz–Simon gradient inequality at a critical point $\overline{u}={\lim }_{n\to \infty }u(t_n)$, where $t_n\to \infty$ and $u=u(t)$ is a precompact solution to the associated gradient flows

$$\{\begin{array}{ccc}{\partial }_{t}u\hfill & =-\mathrm{\nabla }\mathcal{E}(u),\hfill & t>0\hfill \\ u(0)\hfill & =u_0,\hfill & \hfill \end{array}$$

we may conclude that u converges with ${\lim }_{t\to \infty }u(t)=\overline{u}$. Numerical applications of this phenomenon have been considered for instance in [2], [6].

Hence, it is a question of great interest, whether a given energy function satisfies a Łojasiewicz–Simon gradient inequality. It can be shown that in the infinite-dimensional case, mere analyticity of the energy is not enough, see for instance [19, Theorem 2.1, Proposition 3.5]. On the other hand, very general conditions which are sufficient for the gradient inequality to hold are presented in [10].

For most of the applications, one usually checks that the following conditions are satisfied, see [12], [13], [14], [25].

Theorem 1.2 Consequence of [10, Corollary 3.11]

Let V be a Banach space, $U\subset V$ an open set, $\mathcal{E}\in {\mathcal{C}}^{\omega }(U;\mathbb{R})$ and $\overline{u}\in U$ a critical point of $\mathcal{E}$. Suppose that

• there exists a Banach space Z such that $V\hookrightarrow Z$ densely,

• ${\mathcal{E}}^{\prime }\in {\mathcal{C}}^{\omega }(U;Z^{⁎})$,

• the second derivative ${\mathcal{E}}^{″}(\overline{u}):V\to Z^{⁎}$ is Fredholm of index zero.

Then, there exist $C,\sigma >0$, $\theta \in (0,\frac{1}{2}]$ such that for all $u\in U$ with ${\Vert u-\overline{u}\Vert }_V\le \sigma$, we have

$${|\mathcal{E}(u)-\mathcal{E}(\overline{u})|}^{1-\theta }\le C{\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{Z^{⁎}}.$$

Remark 1.3

Note that by assumption (i) in Theorem 1.2 we have $V\hookrightarrow Z$, so $Z^{⁎}$ can be identified with a subset of $V^{⁎}$. Condition (ii) requires that for all $u\in U$ the functional ${\mathcal{E}}^{\prime }(u)$ which is in general only in $V^{⁎}$ is in fact in $Z^{⁎}$ and the map ${\mathcal{E}}^{\prime }:U\to Z^{⁎}$ is analytic.

Although Theorem 1.2 describes a slightly less general situation than in [10], in most applications its conditions are relatively easy to check and suffice to prove the Łojasiewicz–Simon gradient inequality. The details on how to deduce Theorem 1.2 from [10] are given in Appendix A.

To prove a suitable version of Theorem 1.1 on a finite-dimensional manifold $\mathcal{M}$ is quite straightforward if $\mathcal{M}$ and $\mathcal{E}$ are analytic, by simply choosing local coordinates and applying Theorem 1.1. In [23], this is used to study gradient-like dynamical systems via the Kurdyka–Łojasiewicz inequality. The infinite-dimensional setting is more complicated.

Our main result is to extend Theorem 1.2 to a constrained energy function $\mathcal{E}{|}_{\mathcal{M}}$ on a submanifold $\mathcal{M}$ of a Banach space V, and to refine the estimate by projecting the derivative onto the cotangent space of $\mathcal{M}$. In [25], a special case has been studied and a Łojasiewicz–Simon gradient inequality is proven for the Canham–Helfrich energy on the submanifold of closed embedded surfaces with fixed area and volume, see [25, Theorem 1.4]. In the following theorem, we give very general sufficient conditions for the Łojasiewciz–Simon gradient inequality to hold on an infinite-dimensional submanifold in the abstract setting of an energy on a Banach space. In Section 5, we will consider the easier case where the ambient space is a Hilbert space. However, as we shall explain in detail in Remark 1.7 below, in order to avoid issues with analyticity, it is sometimes necessary to work in Banach spaces, cf. also Section 7.1. Our main result is the following

Theorem 1.4

Let V be a Banach space, $U\subset V$ an open set, $m\in \mathbb{N}$ and $\mathcal{E}:U\to \mathbb{R}$, $\mathcal{G}:U\to {\mathbb{R}}^m$ be analytic. Let $\overline{u}\in U$ and suppose that

• there exists a Banach space Y such that $V\hookrightarrow Y$ densely,

• ${\mathcal{E}}^{\prime }\in {\mathcal{C}}^{\omega }(U;Y^{⁎})$,

• the second derivative ${\mathcal{E}}^{″}(\overline{u}):V\to Y^{⁎}$ is Fredholm of index zero,

• for any $u\in U$, the linear operator ${\mathcal{G}}^{\prime }(u)\in \mathcal{L}(V,{\mathbb{R}}^m)$ extends to $\overline{{\mathcal{G}}^{\prime }(u)}\in \mathcal{L}(Y,{\mathbb{R}}^m)$ and the map $\overline{{\mathcal{G}}^{\prime }}:U\to \mathcal{L}(Y,{\mathbb{R}}^m)$, $u\mapsto \overline{{\mathcal{G}}^{\prime }(u)}$ is analytic,

• the Fréchet derivative ${\left(\overline{{\mathcal{G}}^{\prime }}\right)}^{\prime }(\overline{u}):V\to \mathcal{L}(Y,{\mathbb{R}}^m)$ is compact,

• $\mathcal{G}(\overline{u})=0$ and ${\mathcal{G}}^{\prime }(\overline{u}):V\to {\mathbb{R}}^m$ is surjective.

Then, $\mathcal{M}:=\{u\in U|\mathcal{G}(u)=0\}$ is locally an analytic submanifold of V of codimension m near $\overline{u}$.

If $\overline{u}$ is a critical point of $\mathcal{E}{|}_{\mathcal{M}}$, then the restriction satisfies a refined Łojasiewicz–Simon gradient inequality at $\overline{u}$, i.e. there exist $C,\sigma >0$ and $\theta \in (0,\frac{1}{2}]$ such that for any $u\in \mathcal{M}$ with ${\Vert u-\overline{u}\Vert }_V\le \sigma$, we have

$${|\mathcal{E}(u)-\mathcal{E}(\overline{u})|}^{1-\theta }\le C{\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{{\overline{{\mathcal{T}}_u\mathcal{M}}}^{⁎}}.$$

Here, ${\overline{{\mathcal{T}}_u\mathcal{M}}}^{⁎}$ is the dual of the closure $\overline{{\mathcal{T}}_u\mathcal{M}}:={\overline{{\mathcal{T}}_u\mathcal{M}}}^{{\Vert \cdot \Vert }_Y}\subset Y$ of the tangent space ${\mathcal{T}}_u\mathcal{M}$.

Remark 1.5

The notation $\overline{{\mathcal{G}}^{\prime }(u)}$ is justified, since the operator $\overline{{\mathcal{G}}^{\prime }(u)}:Y\to {\mathbb{R}}^m$ is the closure of $A={\mathcal{G}}^{\prime }(u)$ on the Banach space Y with $D(A)=V$.

Remark 1.6

• Note that we could apply Theorem 1.2 in the situation of Theorem 1.4 as well, but (3) yields a sharper estimate: If $Z=Y$ with Y as in Theorem 1.4, then for $u,\overline{u}\in \mathcal{M}$ with ${\Vert u-\overline{u}\Vert }_V\le \sigma$, we have

  $${\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{{\overline{{\mathcal{T}}_u\mathcal{M}}}^{⁎}}=\underset{0\ne y\in \overline{{\mathcal{T}}_u\mathcal{M}}}{\sup }\frac{{\mathcal{E}}^{\prime }(u)y}{{\Vert y\Vert }_Y}\le \underset{0\ne y\in \overline{{\mathcal{T}}_u\mathcal{M}}}{\sup }\frac{{\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{Y^{⁎}}{\Vert y\Vert }_Y}{{\Vert y\Vert }_Y}={\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{Y^{⁎}}.$$

  Thus, if the assumptions of Theorem 1.4 are satisfied and $C,\sigma ,\theta$ are as in Theorem 1.4, we have ${|\mathcal{E}(u)-\mathcal{E}(\overline{u})|}^{1-\theta }\le C{\Vert {\mathcal{E}}^{\prime }(u)\Vert }_{Y^{⁎}}$, i.e. (3) implies (2) under the assumptions of Theorem 1.4. It hence makes sense to refer to (3) as a refined Łojasiewicz–Simon gradient inequality.

• From our proof, we cannot conclude that the Łojasiewicz exponents θ in Theorem 1.2 and Theorem 1.4 coincide.

Remark 1.7

The Hilbert space case treated in Corollary 5.2 is much easier to handle than Theorem 1.4. It is also more natural since one usually studies H-gradient flows with $H=W^{k,2}(\mathrm{\Omega })$, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ open, $k\in \mathbb{Z}$. On the other hand, one may sometimes encounter a problem in proving analyticity of the energy. The problematic phenomenon is, that whenever a Nemytskii or supercomposition operator

$$\mathcal{F}:L^p(\mathrm{\Omega })\to L^q(\mathrm{\Omega }),\mathcal{F}(v)=f(v)=f\circ v\text{ with }p,q\in [1,\infty )$$

is analytic, the function f has to be a polynomial of degree at most $\lceil \frac{p}{q}\rceil$, see [5, Theorem 3.16]. A way to work around this, is to choose suitable Sobolev spaces, such that all derivatives in the energy either appear in polynomial expressions with appropriate powers or are continuous. This is exactly why we work in the Banach space $W^{2,p}(\mathrm{\Omega })$ with $p>d$ to prove the Łojasiewicz–Simon gradient inequality in Section 7.1.

This article is structured as follows. First, we recall some basic definitions and fundamental properties of analytic functions and Fredholm operators. Then we present the generalizations of basic concepts of differential geometry to submanifolds of a Banach space. In Section 3, we establish a local graph representation for the manifold $\mathcal{M}$ in Theorem 1.4. It turns out that studying this chart plays a crucial role in the proof of Theorem 1.4 which we complete in Section 4. After that, we consider the Hilbert space case in Section 5 in which the inequality takes a more convenient form. We also prove an abstract convergence result for the associated gradient flow in this case. Section 6 is dedicated to discuss the necessity of the assumptions we make in Theorem 1.4. In the last section, we will then apply our abstract results to the area of graph surfaces with an isoperimetric constraint in Section 7.1, the Allen–Cahn equation in Section 7.2 and to surfaces of revolution with prescribed volume in Section 7.3.