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).
