Ricerc@Sapienza

We study the asymptotic behavior of anomalous fractional diffusion processes in bad domains via
the convergence of the associated energy forms.We introduce the associated Robin–Venttsel’ problems for
the regional fractional Laplacian. We provide a suitable notion of fractional normal derivative on irregular
sets via a fractional Green formula as well as existence and uniqueness results for the solution of the
Robin–Venttsel’ problem by a semigroup approach. Submarkovianity and ultracontractivity properties of
the associated semigroup are proved.

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.