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Fractal snowflake domain diffusion with boundary and interior drifts. (English) Zbl 1516.35220

Summary: 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.

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35D30 Weak solutions to PDEs
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