Characterization of subdifferentials of a singular convex functional in Sobolev spaces of order minus one (Q765909)

The author analyzes the subdifferential formulation of the \(4\)th order equation \[ \frac\partial{\partial t}f=-\Delta\text{div}(|\nabla f|^{-1}\nabla f+\mu|\nabla f|^{p-2}\nabla f),\;\mu>0,\;p\in (1,\infty), \] where \(f\) is a time-dependent real-value function defined on a bounded domain \(\varOmega\subset \mathbb{R}^d\) obeying an appropriate boundary condition. The equation can be expressed in a gradient flow form \(\frac{\partial f}{\partial t}=-\frac {\delta F(f)}{\delta f}\) governed by the energy functional \[ F(f)=\int_\varOmega \Big (|\nabla f({\mathbf x})|+\frac \mu p |\nabla f({\mathbf x})|^p\Big)\,d{\mathbf x}. \] The functional derivative of \(F\) is taken with respect to the metric in the space \(H^{-1}(\varOmega)\). Both the periodic and a Dirichlet boundary condition are separately imposed.