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

Journal of Functional Analysis

2012

Self Cite

121

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including nonself-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

Section: Introductionmentioning

confidence: 71%

Derivations and Dirichlet forms on fractals

Ionescu

Rogers

Journal of Functional Analysis

2012

Self Cite

121

“…Choosing n larger and larger, we can approximate a given

C^{1} false(R^{2} false)

‐function on S in energy by functions that are linear on these triangles. Given such a piecewise linear function g we can then cover S by

M = M (n^{'})

sets

\bar{O_{i}} = S_{n^{'}, k}

and apply a slight modification of the proof of Theorem to see that if

n^{'}

is chosen large enough, we can find a function

\tilde{g} \in {scriptS}^{⋃_{i = 1}^{M} \partial O_{i}}

that is arbitrarily close to g in energy. (ii)For any compact topologically one‐dimensional metric space X that satisfies Assumption the locally harmonic 1‐forms are dense in the orthogonal complement

{scriptH}^{1} false(X false)

I m \partial

scriptH

, see [, Theorem 4.2]. According to (i) above, this result holds in particular for the generalized Sierpinski carpets S satisfying .…”

Section: Local Dirichlet Forms On Carpet‐like Spacesmentioning

confidence: 95%

“…We consider the Hilbert space

(H, {⟨ \cdot, \cdot ⟩}_{scriptH})

of L 2 ‐ differential 1‐forms associated with

(E, F)

and the corresponding first order derivation

\partial_{0} : A \to H

as introduced in and studied in . This is a generalized L 2 ‐theory of 1‐forms.…”

Section: Local Dirichlet Forms On Carpet‐like Spacesmentioning

confidence: 99%

“…We consider the space

\bar{scriptA} \otimes A

spanned by tensors

f \otimes g

, by

(f \otimes g) (x, y) = f (x) g (y)

they may be viewed as elements of

C (X \times X)

. In order to follow the notation of , the notation used here deviates slightly from the one in [, Section 8.1], but structurally the definitions are the same. Right and left actions of

scriptA

\bar{scriptA} \otimes A

can be defined as the linear extensions of

(f \otimes g) h : = f \otimes (g h) and h (f \otimes g) : = (f h) \otimes g - h \otimes (f g) .

The definition

\partial_{0} f : = f \otimes 1, f \in A,

yields a linear operator

\partial_{0} : A \to \bar{scriptA} \otimes A

that satisfies a product rule ,

\partial_{0} false(f g false) = f \partial_{0} g + false(\partial_{0} f false) g, f, g \in A .

The space

\bar{scriptA} \otimes \bar{scriptA} \otimes ...

…”

Section: Algebraic Definitions Energy Norms and Wedge Productsmentioning

confidence: 99%

“…In particular, applying the (classical) exterior derivation to smooth 1‐forms on

R^{2}

and restricting the resulting 2‐forms to S , we can observe the existence of nonzero square integrable 2‐forms on S . On the other hand, if the holes in the carpet are not too small, then [, Theorem 4.2] applies (see Remark below) and tells that within the L 2 ‐space of 1‐forms the locally harmonic 1‐forms are dense in the orthogonal complement of the exact 1‐forms. Here we call a square integrable 1‐form ω locally harmonic if we can find a finite open cover

{false{U_{α} false}}_{α \in J}

of S and functions

h_{α}

α \in J

, harmonic in the Dirichlet form sense, such that for any

α \in J

we have

ω {bold1}_{U_{α}} = d h_{α} {bold1}_{U_{α}}

, where d denotes the exterior derivation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Densely defined non‐closable curl on carpet‐like metric measure spaces

Hinz

Mathematische Nachrichten

2018

Self Cite

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.

Finite Energy Coordinates and Vector Analysis on Fractals

Hinz

Fractal Geometry and Stochastics V

2015

Self Cite

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.