Lojasiewicz inequality
2020 Mathematics Subject Classification: Primary: 14P15 Secondary: 14P05 [MSN][ZBL]
An inequality on real analytic functions proved by S. Lojasiewicz in [Lo] (see also Liouville-Lojasiewicz inequality). One form of the inequality states the following.
Theorem 1 Consider an open set $U\subset \mathbb R^n$ and an analytic function $f:U \to \mathbb R$. Let $Z_f$ denote the zeros of $f$. Then for every compact set $K\subset U$ there are positive constants $\alpha$ and $C$ such that \begin{equation}\label{e:Loj1} {\rm dist}\, (x, Z_f)^\alpha \leq C |f(x)| \qquad \qquad \forall x \in K. \end{equation}
A different form of the inequality is often used in analysis and is also called Lojasiewicz inequality (sometimes Lojasiewicz gradient inequality)
Theorem 2 Let $U$ and $f$ be as above. For every critical point $x\in U$ of $f$ there is a neighborhood $V$ of $x$, an exponent $\theta\in [\frac{1}{2},1[$ and a constant $C$ such that \begin{equation}\label{e:Loj2} |f(x)-f(y)|^\theta \leq C |\nabla f (y)| \qquad \qquad \forall y\in V\, . \end{equation}
Comments
Observe that
- If we know \eqref{e:Loj2}, we can remove the constant $C$ by restricting the inequality to a smaller neighborhood $W$ of $x$ and substituing $\theta$ with a larger exponent. However, the inequality is usually stated as above since in several applications it is convenient to have the smallest possible exponent $\theta$.
- For $n=1$ it is very easy to prove both \eqref{e:Loj1} and \eqref{e:Loj2} using the power expansion of $f$. More precisely, if the function is not constant, then there is a smallest $N\geq 1$ such that $f^{(N)} (x)\neq 0$. We can then write
\[ f(y) - f(x) = \sum_{k\geq N} \frac{f^{(k)} (x)}{k!} (y-x)^k \] for $y$ in a neighborhood of $x$. Choosing $r$ sufficiently small we easily conclude \[ 2 \left|\frac{f^{(N)} (x)}{N!}\right| |y-x|^N\geq |f(y) - f(x)| \geq \frac{1}{2}\left|\frac{f^{(N)} (x)}{N!}\right| |y-x|^N \qquad \forall y\in ]x-r, x+r[\, . \] Next, assuming that $f' (x) = 0$ (i.e. $N>1$) we also conclude, for $r$ sufficiently small \[ |f'(y)|\geq \frac{1}{2} \left|\frac{f^{(N)} (x)}{(N-1)!}\right| |y-x|^{N-1} \] and we conclude \eqref{e:Loj2} with \[ \theta = \frac{N-1}{N} \] The inequalities are however rather difficult to prove when $n>1$.
- The classical example of a $C^\infty$ function which is not real analytic shows that the analyticity assumption is necessary. Indeed consider
\[ f(x) =\left\{ \begin{array}{ll} e^{-\frac{1}{|x|}} \qquad&\mbox{for } x\neq 0\\ 0 &\mbox{for } x=0 \end{array}\right. \] It can be easily checked that $f\in C^\infty$ and that $Z_f = \{0\}$. It is also obvious that \eqref{e:Loj1} fails in any neighborhood of $0$ and that Theorem 2 does not hold at $x=0$.
Remark 3 Further important remarks:
- By the argument above we draw the important conclusion that the exponent $\theta$ in \eqref{e:Loj2} is necessarily not smaller than $\frac{1}{2}$.
- The inequality \eqref{e:Loj2} can be shown to hold with $\theta = \frac{1}{2}$ when $D^2 f (x)$ is invertible, i.e. $x$ is a non-degenerate critical point: the inequality is indeed a simple consequence of the Morse lemma and the analyticity assumption is superfluous ($f\in C^2$ is sufficient to apply the Morse lemma).
Applications
The Lojasiewicz inequality has found rather striking applications in the theory of ordinary and partial differential equations, in particular to gradient flows. In a finite-dimensional context, a gradient flow is sometimes called gradient dynamical system and consists of a system of ordinary differential equations of the form \begin{equation}\label{e:gradient_flow} \dot{x} (t) = - \nabla f (x(t))\, . \end{equation} The following lemma of Lojasiewicz (see [Lo2]) then holds
Lemma 4 Assume $U\subset \mathbb R^n$ is open, $f: U \to \mathbb R$ is a $C^2$ function and $x:[0, \infty[\to U$ is a solution of \eqref{e:gradient_flow}. Assume the existence of a sequence $t_n\to \infty$ such that $x (t_n)$ converges to a critical point $\bar{x}$ of $f$ which satisfies \eqref{e:Loj2} for some $\theta \in ]0,1[$ in some neighborhood $V$. Then \[ \lim_{t\to \infty} x(t) = \bar{x}\, . \]
Thus, from Theorem 2 we conclude the following important corollary
Corollary 5 Assume $f:\mathbb R^n \to \mathbb R$ is real analytic and let $x:[0,\infty[\to \mathbb R^n$ be a solution of \eqref{e:gradient_flow}. Then, either \[ \lim_{t\to\infty} |x(t)| = \infty \] or there exists $\bar{x}$ critical point of $f$ such that \[ \lim_{t\to\infty} x(t) = \bar{x}\, . \]
It was realized by L. Simon in his pioneering work [Si] that under appropriate ellipticity assumptions the Lojasiewicz inequality can be extended to an infinite-dimensional setting. He then used it to study
- the asymptotic behavior of solutions to parabolic equations
- the asymptotic behavior of solutions to geometric variational problems near an isolated singular point.
For this reason, suitable infinite-dimensional versions of \eqref{e:Loj2} (which can be reduced to Theorem 2 using a Lyapunov-Schmidt reduction, see [Si2]) are often called Lojasiewicz-Simon inequalities.
Remark 6 Corollary 5 can be shown to fail under the assumption $f\in C^\infty$. The exponent $\theta$ in \eqref{e:Loj2} is related to the rate of convergence of $x(t)$ to $\bar{x}$ in Lemma 4. In particular, if $\theta = \frac{1}{2}$, then it can be shown that \[ |x(t) - \bar{x}|\leq e^{- \bar{C} t}\, \] for some $\bar{C}>0$. If $\theta > \frac{1}{2}$, then it can be shown that \[ |x(t)-\bar{x}| \leq \frac{\bar{C}}{t^\alpha}\, , \] where the epxonent $\alpha$ can be explicitly computed from $\theta$. Cf. [Si2].
References
[Lo] | S. Łojasiewicz, "Ensembles semi-analytiques", preprint IHES (1965) |
[Lo2] | S. Łojasiewicz, "Sur les trajectoires du gradient d’une fonction analytique", Seminari di Geometria, Bologna (1982/83), Universita' degli Studi di Bologna, Bologna (1984), pp. 115–117 |
[Lo3] | S. Łojasiewicz, "Une propriété topologique des sous ensembles analytiques réels", Colloques internationaux du C.N.R.S 117. Les Équations aux Dérivées Partielles, (1963) 87-89 |
[Si] | L. Simon, "Asymptotics for a class of non-linear evolution equations, with applications to geometric problems", Ann. Math. 118 (1983), pp. 525-571 |
[Si2] | L. Simon, "Theorems on regularity and singularity of energy minimizing maps" Lectures in Mathematics. ETH Zürich. Birkhäuser, Basel (1996) |
Lojasiewicz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lojasiewicz_inequality&oldid=51341