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We consider a homogenization problem associated with quasi-crystalline multiple inte- grals of the form uε ∈ Lp(Ω;Rd) 7→ ˆΩ fR x, xε,uε(x) dx, where uε is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, fR, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields uε that are in the kernel of a given constant-coefficient linear partial differential operator, A, that is, Auε = 0. Our results provide a generalization of related ones in the literature concerning the A = curl case to more general differential operators A with constant coefficients, and without coercivity assumptions on the Lagrangian fR.
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Inequivalent representations of the dual space
Abstract. We show that there exist inequivalent representations of the dual space of C[0, 1] and of Lp[Rn] for p ∈ [1, ∞). We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if A is a proper closed subspace of C[0, 1], there always exists a closed subspace A⊥ (with the same definition as for L2[0, 1]) such that A∩A⊥ = {0} and A⊕A⊥ = C[0, 1]. Thus, the geometry of C[0, 1] can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator A on C[0, 1] has a uniquely defined adjoint A∗ defined on C[0, 1], and both can be extended to bounded linear operators on L2[0, 1]. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to Lp[Rn]. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on C[0, 1] is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.
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- Award ID(s):
- 1928930
- PAR ID:
- 10431791
- Date Published:
- Journal Name:
- Revista de la Unión Matemática Argentina
- ISSN:
- 1669-9637
- Page Range / eLocation ID:
- 271 to 280
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.33044/revuma.3065
