Fractal snowflake domain diffusion with boundary and interior drifts

Abstract

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.

Discretization of the Koch Snowflake Domain with Boundary and Interior Energies

“…The Koch snowflake and the associated snowflake domain are well-known. Here we introduce some notation and foundational results for our analysis, following [25]. Let {F i } 4 i=1 be the iterated function system defined on C by…”

“…The main objective of this paper is to investigate a discrete version of the eigenvalue problem ∆u = λu on the Koch snowflake domain, which we denote by Ω. To this end, we follow [25] and introduce a Dirichlet form (with a suitable domain) on Ω,…”

“…The paper is organized as follows. Section 2, follows the treatment in [25] to introduce a Dirichlet form on the Koch snowflake. In Section 3 we give the Dirichlet form on the snowflake domain and discuss some of its properties, such as the Hölder continuity and uniform approximation of eigenfunctions.…”

Fractal Logistic Equation

“…Fractal geometry includes shapes which are scale invariant and have fractional dimensions and self-similar properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Analysis on fractals was formulated using different methods such as harmonic analysis, probabilistic methods, measure theory, fractional calculus, fractional spaces, and time-scale calculus [18][19][20][21][22][23][24][25][26][27][28][29].…”

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

“…One part of this very large literature deals with spaces where Lipchitz functions can be analyzed. Without even attempting to suggest a representative sample of relevant papers, we briefly mention such recent works as [1,10,32,36]. Another kind of more probabilistically inspired analysis deals with spaces that are more fractal in nature and have sub-Gaussian heat kernel estimates (see [2-4, 6, 7, 9] and references therein).…”