Quasi-linear Venttsel problems with nonlocal boundary conditions on fractal domains
Let $\Omega\subseteq\mathbb{R\!}^{\,2}$ be an open domain with fractal boundary $\partial\Omega$. We define a proper, convex and lower semicontinuous functional on the space $\mathbb{X\!}^{\,2}(\Omega,\partial\Omega):=L^2(\Omega,dx)\times L^2(\partial\Omega,d\mu)$, and we characterize its subdifferential, which gives rise to nonlocal Venttsel' boundary conditions. Then we consider the associated nonlinear semigroup $T_p$ generated by the opposite of the subdifferential, and we prove that the corresponding abstract Cauchy problem is uniquely solvable.