Fractal snowflake domain diffusion with boundary and interior drifts

TL;DR

This paper investigates a parabolic differential equation with boundary and interior drifts on a fractal snowflake domain, establishing existence, uniqueness, and Lipschitz extension properties using advanced fractal analysis techniques.

Contribution

It introduces a rigorous framework for solving Ventsell problems on fractal domains, combining PDE analysis with fractal geometry methods.

Findings

Unique classical solution to the parabolic Ventsell problem established.

Lipschitz functions on the snowflake boundary can be extended to the entire domain.

Methodology integrates fractal membrane analysis and PDE techniques on fractals.

Abstract

We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector analysis for local Dirichlet forms and PDE on fractals, coercive closed forms, and the analysis of Lipschitz functions.