Entropy and differential Harnack type formulas for evolving domains (Q2248856)

The author investigates the evolution with normal speed \(\beta=\beta_{M_t}=-\frac{\partial x}{\partial t}\cdot \nu\) of bounded open regions \((\Omega_t)_{t\in (0,T)}\) in \(\mathbb{R}^{n+1}\) with smooth boundary \(M_t=\partial\Omega_t\), where \(x\) denotes the embedding map of \(M_t\) and \(\nu\) is normal to \(M_t\) pointing out of \(\Omega_t\). The author studies the evolution equation \[ \frac{\partial f}{\partial t} + \Delta f = |\nabla f|^2 + \frac{n+1}{2\tau} \] in \(\Omega_t\) with Neumann boundary condition \(\nabla f \cdot \nu = \beta\) under the assumptions \(\tau=\tau(t)>0\) and \(\frac{\partial\tau}{\partial t}=-1\). The main goal of the paper is to show that for some natural speed functions \(\beta\), the right hand side of \[ \begin{aligned} \frac{d}{dt} \mu_{\beta}(\Omega_t,\tau(t)) \geq & \;2\tau(t)\int_{\Omega_t} |\nabla_i\nabla_j f-\frac{\delta_{ij}}{2\tau(t)} |^2 u dx\\ & +2\tau(t) \int_{M_t} \left(\frac{\partial\beta}{\partial t} - 2\nabla^{M_t} \beta\cdot\nabla^{M_t}f + A_{M_t}(\nabla^{M_t} f,\nabla^{M_t} f)-\frac{\beta}{2\tau(t)} \right) u dS,\end{aligned} \] where \(\mu_{\beta}(\Omega_t,\tau(t))\) and \(A_{M_t}\) are the associated entropy and the second fundamental form of \(M_t\), respectively, is non-negative, that is the inequality \(\frac{d}{dt} \mu_{\beta}(\Omega_t,\tau(t))\geq 0\) holds. The author concentrates his attention on the mean curvature flow \(H_{M_t}\) of the boundary hypersurfaces. He presents some Li-Yau-Hamilton type Harnack inequalities for a mean curvature flow consisting of compact hypersurfaces involving a minimizer of the associated entropy \(\mu_{\beta}(\Omega_t,\tau(t))\).