Homogenization in Fractional Elasticity

Abstract

In this note we treat the equations of fractional elasticity. After establishing well-posedness, we show a compactness result related to the theory of homogenization. For this, a previous result in (abstract) homogenization theory of evolutionary equations has to be improved. The approach also permits the consideration of non-local operators (in time and space).

G‐convergence and the weak operator topology

“…In order to sketch an application of Theorem 1.2, we recall a special case of the main result of [4]. For this note that we identify closed operators on a Hilbert space H with its (abstract) multiplication operators on H-valued functions with its maximal domain.…”

On abstract grad–div systems

“…In a number of studies [21,20,14,33,29,22] it has been illustrated, that typical initial boundary value problems of mathematical physics can be represented in the general form…”

Homogenization in fractional elasticity – One spatial dimension

“…For an account of the methods mentioned here we refer to [9] with extensions in [10][11][12]. The equation of (fractional) elasticity can be found in [7,Section 3.1.12.5] or [5, pp.…”

“…102]. A Kelvin-Voigt type model for one spatial dimension (see also [12,Example 2.6]) may be written as follows. For (t, x) ∈ R×]0, 1[ consider…”

“…Assuming periodicity of µ, C and D, we show the weak convergence of (u n , T n ) n and derive an equation satisfied by the respective limit. A more detailed account of the results mentioned can be found in [12]. In particular, we refer to [12, Example 2.6, Example 3.5 and Remark 3.6].…”