Are there Lojasiewicz-Simon estimates with boundary?

Question

Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary.

Are there Lojasiewicz–Simon estimates available for minimal surfaces $\Sigma$ in $M$ with $\partial \Sigma \subset \partial M$? If necessary, assume that $\partial \Sigma$ is analytic also.

I would also be interested in the special case where $M$ is the Euclidean ball $B^n \subset \mathbf{R}^n$.

My naive guess for a formulation in a manifold without boundary, still denoted $M$ would go as follows; this is essentially copied verbatim from Leon Simon. Let $\Sigma \subset M$ be an embedded, smooth minimal surface.

• Let $\mathcal{M}$ be the minimal surface operator over $\Sigma$, and $\mathcal{A}$ the area operator, so that if $u \in C^1(\Sigma)$ has small enough norm, then $\mathcal{A}(u) = \mathcal{H}^n(\operatorname{graph}_{\Sigma} u)$; in particular $\mathcal{A}(0) = \mathcal{H}^n(\Sigma)$.
• Let also $\mu \in (0,1)$ be an arbitrary Hölder exponent, and $\beta > 0$ be a constant small enough in terms of $\Sigma$ to guarantee the ellipticity of $\mathcal{M}$ over $\Sigma$. Finally, let $\mathcal{S} = \{ \zeta \in C^2(\Sigma) \mid \lvert \zeta \rvert_{C^2} < \beta, \mathcal{M}(\zeta) = 0 \}$.

There are constants $\theta \in (0,1/2)$, $\gamma \geq 2$, $\sigma > 0$ so that if $u \in C^{2,\mu}(\Sigma)$ has $\lvert u \rvert_{C^{2,\mu}} < \sigma$ \begin{equation} \tag{1} \lvert \mathcal{M}(u) \rvert_{L^2} \geq (\operatorname{inf}_{\zeta \in \mathcal{S}} \lvert u - \zeta \rvert_{L^2})^\gamma \end{equation} and \begin{equation} \tag{2} \lvert \mathcal{M}(u) \rvert_{L^2} \geq \lvert \mathcal{A}(u) - \mathcal{A}(0) \rvert^{1 - \theta}. \end{equation}