No abstract

We study a quasi-linear evolution equation with nonlinear dynamical boundary conditions in a two dimensional domain with Koch-type fractal boundary. We consider suitable approximating pre-fractal problems in the corresponding pre-fractal varying domains. After proving existence and uniqueness results via standard semigroup approach, we prove that the pre-fractal solutions converge in a suitable sense to the limit fractal one via the Mosco convergence of the energy functionals adapted by Tölle to the nonlinear framework in varying Hilbert spaces.

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains.

We consider a magnetostatic problem in a 3D "cylindrical" domain of Koch type. We prove existence and uniqueness results for both the fractal and prefractal problems and we investigate the convergence of the pre-fractal solutions to the limit fractal one. We consider the numerical approximation of the pre-fractal problems via FEM and we give a priori error estimates. Some numerical simulations are also shown. Our long term motivation includes studying problems that appear in quantum physics in fractal domains.

We study a nonlocal Robin–Venttsel’-type problem for the regional fractional p-Laplacian in an extension domain $$\Omega $$ Ω with boundary a d-set. We prove existence and uniqueness of a strong solution via a semigroup approach. Markovianity and ultracontractivity properties are proved. We then consider the elliptic problem. We prove existence, uniqueness and global boundedness of the weak solution.