Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\ge2$ and $\delta>0$ such that $$(1)~~~\operatorname{dist}(x,V)^{\gamma}\leq \|\nabla F(x)\|$$ $$(2)~~~|F(x)-F(0)|^{1-\theta}\leq \|\nabla F(x)\|$$ for all $x$ with $\|x\|\le \delta.$ Here $V=\{y\in\mathbb{R}^n:\|\nabla F(y)\|=0 \}.$ The constants $\theta,\gamma$ and $\delta$ depend on the functional $F$ at $0.$

An important generalization by L. Simon to a class of analytic functionals on certain Holder spaces is given here in Theorem 3, which has become known as the Lojasiewicz-Simon gradient inequality.

Simon's work: Consider the energy functional $\mathcal{E}(u)=\int_{\Sigma}E(x,u,\nabla u)$ on a $C^{\infty}$ Riemannian manifold $\Sigma$, where $E$ is assumed to have analytics dependence on $u,\nabla u$ for $|u|_{C^1(\Sigma)}$ sufficeintly small, and where $\mathcal{M}(0)=0$ ($\mathcal{E}$ has some more properties by I will ignore them here). Here $\mathcal{M}(u)=-(\operatorname{grad}\mathcal{E}(u))$ is the Euler-Lagrange operator for $\mathcal{E};$ that is, the unique function with the following property $$-(\mathcal{M}(u),\xi)_{L^{2}(\Sigma)}=\frac{d}{ds}\mathcal{E}(u+s\xi)\Big|_{s=0}$$ for all $u,\xi\in C^{2}(\Sigma).$

Theorem 3 There are constants $\theta\in(0,\frac{1}{2}]$, $\gamma\ge 2$ and $\eta,\sigma,\beta$ such that if $u$ is an arbitrary function in $C^{2,\eta}(\Sigma)$ with $\|u\|_{C^{2,\mu}(\Sigma)}<\sigma$ then $$(1)~~~\inf_{\{\xi\in C^{\Sigma}: |\xi\|_{C^2(\Sigma)}<\beta,~ \mathcal{M}(\xi)=0\}}\|u-\xi\|_{L^2(\Sigma)}^{\gamma}\leq \|\mathcal{M}(u)\|_{L^2(\Sigma)}.$$ $$(2)~~~|\mathcal{E}(u)-\mathcal{E}(0)|^{1-\theta}\leq \|\mathcal{M}(u)\|_{L^{2}(\Sigma)}.$$

My Question: Under what conditions $\gamma=2$ and $\theta=1/2?$ I would like to know a criterion that also works for the gradient when it is a fully nonlinear elliptic operator (rather than a quasilinear one).


For Inequality (2) in Simon's Theorem 3, a discussion of when the optimal exponent, $\theta=1/2$, is attained can be found (along with many references) in my paper Lojasieicz-Simon gradient inequalities for analytic and Morse-Bott functionals on Banach spaces and applications to harmonic maps with Manos Maridakis. Briefly, there are essentially three (not necessarily distinct) situations where one has the optimal exponent. The first (and simplest) situation is illustrated by Proposition 1.2 in our paper. The second situation occurs when the potential function $\mathcal{E}$ obeys a type of Morse-Bott condition at the critical point and is illustrated by Theorem 2 in our paper. The third situation occurs when the critical point is `integrable'. The recent paper Slowly converging Yamabe flows by Carlotto, Chodosh, and Rubinstein contains a very thorough analysis and discussion of the integrable case in the setting of Yamabe scalar curvature gradient flow.


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