Local and nonlocal Venttsel' problems in fractal domains

The aim of this dissertation is to study local and nonlocal Venttsel' problems in fractal domains. From the physical point of view, such as e.g. in the framework of heat propagation, it turns out that the boundary or the interface acts as a preferential fast absorbing trail for the heat stream. It could be important for industrial applications to enhance the surface effects with respect to the surrounded volume, by increasing the surface (or the length); hence, fractal boundaries and interfaces turn out to be a good tool. From the mathematical point of view, a Venttsel' problem is described by an evolution equation in the bulk coupled with an evolution equation on the boundary where the operators appearing in the bulk and the boundary equations are of the same order. These boundary conditions, known in literature also as dynamical boundary conditions, are the most general ones in literature, since they include Dirichlet, Neumann and Robin boundary conditions. In view of concrete applications to real world problems, the ambitious aim of this thesis is to study both theoretically and numerically Venttsel' problems in 2D and 3D fractal domains, such as Koch-type domains, which are prototypes of irregular domains. More precisely, in addition to the study of the “continuous” problem at hand, we approximate the fractal domain in terms of the so-called “pre-fractal” domains, which are typically non-convex domains having polygonal boundary. We are also interested in considering the numerical approximation of the pre-fractal problems as well as in carrying out some numerical simulations to interpret the physical results. In Chapter 2, we consider an elliptic equation for the Laplace operator in a two-dimensional piecewise smooth domain coupled with a nonlocal linear Venttsel' boundary condition. We prove existence, uniqueness and regularity results for the weak solution in weighted Sobolev spaces. As to the regularity, the presence of the nonlocal term requires to adopt new tools. The techniques used deeply rely on the fact that the nonlocal term can be regarded as a sort of “regional" fractional Laplacian. We first prove a priori estimates, by means of the so-called “Munchhausen trick”, and only after proving the existence and uniqueness of the weak solution, we prove that it has the desired regularity. In Chapter 3, we consider the numerical approximation of a heat equation with nonlocal Venttsel' boundary conditions in a fixed Koch-type pre-fractal (non-convex) domain. We prove existence and uniqueness of the weak (strict) solution via a semigroup approach. We then perform the numerical approximation by mixed methods: we first approximate the problem in the space variable by using a finite element method and then we use a finite difference scheme in time. In order to obtain a priori error estimates and an optimal rate of convergence, it is crucial to obtain regularity results of the weak solution as well as an ad-hoc mesh (satisfying the so-called Grisvard conditions) refined near the singular points. By following the approach developed by Grisvard for non-convex plane domains in the case of the Laplace operator and by adapting the results of Chapter 2, we prove regularity results of the strict solution in weighted Sobolev spaces. We then approximate the problem also in the time variable and, thanks to the regularity results, we achieve the optimal rate of convergence of our numerical scheme, i.e. we obtain the same rate of convergence as in the case of convex domains as well as in the linear local case. We conclude the chapter by presenting also some preliminary numerical results. These simulations show that the presence of the nonlocal term actually helps the process of heat flow through the boundary. In Chapters 4 and 5, we consider local quasi-linear Venttsel' boundary conditions on fractal (and pre-fractal) sets in the 2D and 3D case respectively, involving a “fractal” p-Laplace Beltrami-type operator on the boundary. The focus ok these Chapters is to prove the convergence (in a suitable sense) of the pre-fractal solutions to the limit fractal one. This convergence is achieved by means of the M-convergence of energy functionals, introduced by Mosco and generalized to our setting firstly by Kuwae and Shioya and then by Tölle. In Chapter 4, we consider a bounded “fractal” domain with boundary the Koch snowflake and the natural corresponding approximating “pre-fractal” domains. We introduce two suitable nonlinear energy functionals in the fractal and pre-fractal case. The pre-fractal energy functionals are defined on suitable varying Hilbert spaces, which converge (in a suitable sense) to another Hilbert space, which is the domain of the fractal energy functional. These functionals are the sum of a nonlinear p-energy term in the bulk, a nonlinear fractal p-energy term on the boundary plus lower order terms and are proper, lower semicontinuous and convex on their domains. By a nonlinear semigroup approach, we get existence and uniqueness of the “strong” solution of the corresponding abstract homogeneous Cauchy problems, involving the subdifferentials of the nonlinear energy functionals. At last, we prove that the solutions of the abstract Cauchy problems solve, in a suitable sense, quasi-linear PDEs with quasi-linear local Venttsel' boundary conditions, and that the solutions of the pre-fractal problems converge to the solution of the fractal problem. In Chapter 5, we consider the quasi-linear local Venttsel' problem in a three-dimensional “fractal” cylindrical domain having as lateral surface the Cartesian product between the Koch curve and the unit interval, and also the natural corresponding approximating “pre-fractal” domains. We try to adapt the results of Chapter 4 to the 3D case. We prove new density results in order to suitably approximate the functions in the domains of the energy functionals in terms of smoother functions. We then consider the quasi-linear evolution Cauchy problems with nonlocal Venttsel' boundary conditions both in the fractal and pre-fractal case. We prove existence and uniqueness results by nonlinear semigroup theory. As in the 2D case, the M-convergence of the functionals allows us to prove (in a suitable sense) the convergence of the solutions of the pre-fractal problems to the limit fractal one.