Energy Form on a Closed Fractal Curve

Abstract

The energy form on a closed fractal curve F is constructed. As F is neither self-similar nor nested, it is regarded as a "fractal manifold". The energy is obtained by integrating the Lagrangian on F .

A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences

“…It can be proved as in Proposition 3.1 of [22], that: .2). For the definition and properties of regular Dirichlet forms we refer to [27].…”

“…It is known that the limit (2.3) exists at quasi every P ∈  with respect to the ( ) See [22] and [25]. Throughout the paper c will denote possibly different constants.…”

Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

“…Let T ⊂ R be a closed nonempty subset. It is a -set (0 < ≤ ) if there exists a Borel measure with supp = T such that, for some constants 1 = 1 (T) > 0 and 2 = 2 (T) > 0, See [23,26]. We now come to the definition of the Besov spaces.…”

“…Note that in the case of the Koch curve, the Lagrangian is absolutely continuous with respect to the measure ; on the contrary, this is not true on most fractals (see [24]). In [23] the Lagrangian L on the snowflake has been defined by using its representation as a fractal manifold. Here we do not give details on the construction and definition of L and we refer to Section 4 in [23] for details; in particular in Definition 4.5 a Lagrangian measure L on and the corresponding energy form E as…”

“…In [23] the Lagrangian L on the snowflake has been defined by using its representation as a fractal manifold. Here we do not give details on the construction and definition of L and we refer to Section 4 in [23] for details; in particular in Definition 4.5 a Lagrangian measure L on and the corresponding energy form E as…”