Measurable Riemannian structures associated with strong local Dirichlet forms

Lancia

Journal of Mathematical Analysis and Applications

et al. 2018

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.

Section: Tangential Gradients On the Snowflakementioning

confidence: 99%

“…By Cauchy-Schwarz there is a constant c > 0 such that for any g ∈ V (Ω) and v ∈ H 1 (Ω) we have (21) |l…”

Section: Strong Interpretation and Co-normal Derivativesmentioning

confidence: 99%

Fractal snowflake domain diffusion with boundary and interior drifts

Lancia

Journal of Mathematical Analysis and Applications

et al. 2018

“…This identity expresses the energy in terms of coordinates. As the matrix Z varies measurably in x, it has been named a measurable Riemannian metric, [14,30,34]. The following is version of (6) immediately following from the chain rule [13, Theorem 3.3.2].…”

Section: Energy Fibers and Bundlesmentioning

confidence: 99%

“…∂F ∂y n i ∂y n i Examples 5.1. In the Euclidean and Riemannian situations (1) and (2) in Examples 3.1 the operator ∂ may be identified with the exterior derivation and formula (14) becomes the classical identity in (9).…”

Section: Differential and Gradient In Coordinatesmentioning

confidence: 99%

“…For a general metric measure space we can not expect to find local coordinates that transform smoothly. However, in the field of analysis on fractals Kusuoka [35,36], Kigami [32,34], Strichartz [42], Teplyaev [44], Hino [14], Kajino [30,31] and others have contributed to a concept that is now referred to as 'measurable Riemannian geometry'. This concept is based on Dirichlet forms and involves the use of harmonic functions as 'global coordinates'.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Finite Energy Coordinates and Vector Analysis on Fractals

Fractal Geometry and Stochastics V

2015

Abstract. We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.

Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals

Trans. Amer. Math. Soc.

2014

Abstract. We consider finite energy and L 2 differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion harmonicity for finite energy 1-forms. As as corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classicalČech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence-free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions.