Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems

DOI10.1090/S0002-9947-2010-05143-8zbMath1250.46030arXiv0707.1104MaRDI QIDQ2997228

Claudia Garetto, Hans Vernaeve

Publication date: 6 May 2011

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0707.1104

zbMATH Keywords

variational problemColombeau algebraHilbert moduletopological moduleLax-Milgram theorem

Mathematics Subject Classification ID

Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) (46F30)

Related Items (13)

Optimal embeddings of ultradistributions into differential algebras ⋮ New topologies on Colombeau generalized numbers and the Fermat-Reyes theorem ⋮ Extensions of the Lax-Milgram theorem to Hilbert \(C^*\)-modules ⋮ The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data ⋮ Regularity properties of distributions through sequences of functions ⋮ A Convenient Notion of Compact Set for Generalized Functions ⋮ Strongly internal sets and generalized smooth functions ⋮ Foundations of the calculus of variations in generalized function algebras ⋮ Classes of Generalized Functions with Finite Type Regularities ⋮ Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data ⋮ Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers ⋮ Algebras of generalized functions with smooth parameter dependence ⋮ Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

Cites Work

• Microlocal analysis in the dual of a Colombeau algebra: generalized wave front sets and noncharacteristic regularity
• A global theory of algebras of generalized functions
• First-order hyperbolic pseudodifferential equations with generalized symbols
• Internal sets and internal functions in Colombeau theory
• Topological structures in Colombeau algebras: Topological \(\widetilde{\mathbb C}\)-modules and duality theory
• Topological structures in Colombeau algebras: Investigation of the duals of \(\mathcal G_c(\Omega)\), \(\mathcal G(\Omega)\) and \(\mathcal G_{\mathcal S}(\mathbb R^n)\)
• On the foundations of nonlinear generalized functions I and II
• Generalized pseudo-Riemannian geometry
• Closed graph and open mapping theorems for topological $ \tilde \lc $-modules and applications
• Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions
• Ideals in the Ring of Colombeau Generalized Numbers
• Some remarks on functoriality of Colombeau's construction; topological and microlocal aspects and applications
• Generalized function algebras as sequence space algebras
• Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients
• Microlocal analysis and global solutions of some hyperbolic equations with discontinuous coefficients
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