The Laplace operator on the Sierpinski gasket with Robin boundary conditions

TL;DR

This paper investigates the Laplace operator on the Sierpinski gasket with nonlinear Robin boundary conditions, demonstrating the generation of a positive semigroup and characterizing all local semigroups between Dirichlet and Neumann cases.

Contribution

It provides a novel analysis of Robin boundary conditions on fractal domains, establishing semigroup domination and characterizing generators in this context.

Findings

The Laplace operator with certain Robin conditions generates a positive, order-preserving, contractive semigroup.

The semigroup is dominated by Dirichlet and Neumann Laplacian semigroups.

All local semigroups between these extremals are generated by Robin-Laplace operators.

Abstract

We study the Laplace operator on the Sierpinski gasket with nonlinear Robin boundary conditions. We show that for certain Robin boundary conditions the Laplace operator generates a positive, order preserving, $L^\infty$-contractive semigroup which is sandwiched (in the sense of domination) between the semigroups generated by the Dirichlet-Laplace operator and the Neumann-Laplace operator. We also characterise all local semigroups which are sandwiched between these two extremal semigroups by showing that their generators are Robin-Laplace operators.