Abstract
Generalized Functions are crucial in the development of theories modeling physical reality. Starting with the linear theory, we give a partial account of what is known, focussing on the non-linear theory and some of its more recent achievements.
Similar content being viewed by others
References
Schwartz, L.: Sur l’impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 239, 847–848 (1954)
Schwartz, L.: Théorie des distributions , Publications de l’Institut de Mathématique de l’Université de Strasbourg, No.IX-X, Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris (1966)
Hormander, L.: Linear Partial Differential Operators. Springer, New York (1976)
Hormander, L.: The analysis of linear partial differential operators, 2nd edn. Springer, Berlin (1990)
Lewy, H.: An example of a smooth linear partial differential equation without solution. Ann. Math. 66(2), 155–158 (1957)
Nirenberg, L., Treves, F.: Solvability of a first order linear partial differential equation. Commun. Pure Appl. Math. 16, 331–351 (1963)
Nachbin, L.: Recent developments in infinite dimensional holomorphy. Bull. AMS 79, 625–640 (1973)
Nachbin, L.: Some holomorphically significant properties of ovally convex spaces, J. Funct. Anal. 251–277 (1976)
Rosinger, E.E.: Nonlinear partial differential equations. Sequential and weak solutions, North Holland, Amsterdam (1980)
Rosinger, E.E.: Generalized solutions of nonlinear partial differential equations. North Holland, Amsterdam (1987)
Sebastiãoe Silva, J.: Sui fondamenti della teoria dei funzionali analitici. Port. Math. 12, 1–46 (1953)
Sebastião e Silva, J.: Sur un construction axiomatique de la théorie des distributions, Revista da Faculdade de Ciencias de Lisboa. II Ser. A4. 79-186 (1955)
Kunzinger, M.: Lie transformation group in Colombeau’s algebras. Doctoral Thesis, University of Viena, (1996)
Colombeau, J.F.: Differential Calculus and Holomorphy. Real and Complex Analysis in Locally Convex Spaces, North Holland, Amsterdam (1982)
Colombeau, J.F.: New generalized functions and multiplication of distributions. North Holland, Amsterdam (1984)
Colombeau, J.F.: New Generalized functions multiplication of distributions. Phys. Appl. Port. Math. 41(1–4), 57–69 (1982)
Colombeau, J.F.: Elementary introduction to new generalized functions. North Holland, Amsterdam (1985)
Colombeau, J.F.: Multiplication de distributions et acoustique. J. Acoust. 1, 9–14 (1988)
Colombeau, J.F., Langlais, M.: Generalized solutions of nonlinear parabolic equations with distributions as initial conditions. J. Math. Anal. Appl. 145(1), 186–196 (1990)
Aragona, J., Biagioni, H.: Intrinsic definition of the colombeau algebra of generalized functions. Anal. Math. 17, 75–132 (1991)
Aragona, J., Fernandez, R., Juriaans, S.O., Oberguggenberger, M.: Differential calculus and integration on generalized functions over membranes. Monatsh. Math. 166, 1–18 (2012)
Aragona, J., Fernandez, R., Juriaans, S.O.: A discontinuous Colombeau differential calculus. Monatsh. Math. 144(10), 13–29 (2005)
Aragona, J., Garcia, A.R.G., Juriaans, S.O.: Algebraic theory of Colombeau’s generalized numbers. J. Algebra 384, 194–211 (2013)
Aragona, J., Fernandez, R., Juriaans, S.O.: Natural Topologies on colombeau algebras. Topol. Methods. Nonlinear Anal. 34(1), 161–180 (2009)
Aragona, J., Juriaans, S.O., Oliveira, O.R.B., Scarpalézos, D.: Algebraic and geometry theory of the topological ring of Colombeau generalized functions. Proc. Edinb. Math. Soc. 51(3), 545–564 (2008)
Aragona, J., Juriaans, S.O.: Some structural properties of the topological ring of Colombeau generalized numbers. Commun. Alg. 29(5), 2201–2230 (2001)
Aragona, J., Soares, M.: An existence theorem for an analytic first order PDE in the framework of Colombeau’s theory. Monatsh. Math. 134, 9–17 (2001)
Scarpalezos, D.: Topologies dans les espaces de nouvelles fonctions generalisées de Colombeau. \({\overline{\mathbb{C}}}\) topologiques, Université Paris 7 (1993)
Scarpalezos, D.: Colombeau’s generalized functions: topological structures micro local properties. A simplified point of view, CNRS-URA212, Université Paris 7 (1993)
Scarpalezos, D.: Colombeau’s generalized functions: topological structures; microlocal properties—a simplified point of view part I. Bull. Sci. Math. Nat. Sci. Math. 121(25), 89–114 (2000)
Grosser, M., Kunzinger, M., Steinbauer, R., Oberguggenberger, M.: Geometric theory of generalized functions with application to general relativity. Kluwer Acad Publ, Boston (2001)
Grosser, M., Kunzinger, M., Steinbauer, R., Vickers, J.A.: A global theory of algebras of generalized functions. Adv. Math. 166(1), 50–72 (2002)
Alvarez, A.C., Meril, A., Valiño-Alonso, B.: Step soliton generalized solutions of the shallow water equations, Hindawi Publishing Corporation. J. Appl. Math. (2012)
Aragona, J., Colombeau, J.F., Juriaans, S.O.: Locally convex topological algebras of generalized functions: compactness and nuclearity in a nonlinear context. Tran. Am. Math. Soc. 367, 5399–5414 (2015)
Aragona, J., Colombeau, J.F., Juriaans, S.O.: Nonlinear generalized functions and jump conditions for a standard one pressure liquid-gas model. J. Math. Anal. Appl. 418, 964–977 (2014)
Aragona, J., Garcia, R.G., Juriaans, S.O.: Generalized solutions of a nonlinear parabolic equations with generalized functions as initial data. Nonlinear Anal. 71, 5187–5207 (2009)
Aragona, J., Villarreal, F.: Colombeau’s theory and shock waves in a problem of hydrodynamics. J. D’Analyse Math. 61, 113–144 (1993)
Brézis, H., Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. 62, 73–97 (1983)
Garetto, C.: Topological structures in Colombeau algebras I: topological \({\overline{\mathbb{C}}}\)-modules and duality theory. arXic:math.GN/0407015V1
Garetto, C.: Pseudodifferential operators with generalized symbols and regularity theory. Electron. J. Diff. Equ. 116, 1–43 (2005)
Oberguggenberger, M.: Multiplication of distributions and application to partial differential equations 259, Longman scientific and techinical harlow (1992)
Kamleh, W.: Signature changing space-times and the new generalized functions. preprint
Kunzinger, M., Steinbauer R.: Generalized psuedo-riemannian geometry. Trans. Amer. Math. Soc. 354(10), 4179–4199
Prusa, V., Řehoř, M., Tuma, K.: Colombeau Algebra as mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots, Zeitschrift für angewandte Mathematik und Physik ZAMP. (2016)
Bosch, S., Güntzer, U., Remmert, R.: Non-archimedean analysis. Springer, Berlin, Heidelberg (1984)
Garcia, A.R.G., Juriaans, S.O., Rodrigues, W., Silva, J.C.: Off diagonal condition. preprint
Todorov, T.D., Vernaeve, H.: Full algebra of generalized functions and non-standard asymptotic analysis. Log. Anal. 205–234, (2008)
Gillman, L., Jerison, M.: Rings of continuous functions. Van Nostrand, New York (1960)
Khelif, A., Scarpalezos, D.: Maximal closed ideals of Colombeau algebras of generalized functions. Submitted
Aragona, J.: Colombeau generalized functions on quasi-regular sets. Publ. Math. 68(3–4), 371–399 (2006)
Aragona, J.: On existence theorems for the del-bar operator on generalized differential forms. Proc. Lond. Math. Soc. 53(3), 474–488 (1986)
Aragona, J.: Some results for the \({\overline{\partial }}\) operator on generalized differential forms. J. Math. Anal. Appl. 180, 458–468 (1993)
Oberguggenberger, M., Pilipovic, S., Valmorin, V.: Global representatives of colombeau holomorphic generalized functions. Monatshefte für Math. 151, 67–74 (2007)
Oberguggenberger, M., Vernaeve, H.: Internal sets and internal functions in Colombeau theory. J. Math. Anal. Appl. 341, 649–659 (2008)
Colombeau, J.F., Galé, J.E.: Analytic continuation of generalized functions. Acta Math. Hung. 52, 57–60 (1988)
Colombeau, J.F., Galé, J.: Holomorphic generalized functions. J. Math. Anal. Appl. 103, 117–133 (1984)
Khelif, A.: Scarpalezos, D.: Zeros of generalized holomorphic functions. Monatschefte fur Math. 325–333 (2006)
Garcia, A.R.G., Juriaans, S.O., Rodrigues, W.M., Silva, J.C.: Sets of uniqueness for holomorphic nets. Submitted
Cortes, W., Ferrero, M., Juriaans, S.O.: The Colombeau quaternion algebra. Contemp. Math. 499, 37–45 (2009)
Cortes, W., Garcia, A.R.G., Da Silva, S.H.: The ring of Colombeau’s full generalized quaternions. Commun. Algebra 45, 1532–4125 (2017)
Oberguggenberger, M.: Multiplication of distributions and applications to partial differential equations. Pitman (1992)
Oberguggenberger, M., Pilipovic, S., Scarpalezos, D.: Local properties of Colombeau generalized functions. Preprint (2001)
Pilipovic, S.: Analytic, real analytic and Harmonic generalized functions. Saravejo J. Math. 7, 207–222 (2011)
Todorov, T.D.: Pointwise value and fundamental theorem in the algebra of asymptotic functions. arXiv (2006)
Vernaeve, H.: Ideals in the ring of Colombeau generalized numbers. Commun. algebra 38(6), 2199–2228 (2010)
Kunzinger, M.M.: Characterization of Colombeau generalized functions by their point value. Math. Nachr. 203, 147–157 (1999)
Hörmann, G.: The Cauchy problem for Schrödinger-type partial differential operators with generalized functions in the principal part and as data. Monatsh. Math. 163, 445–460 (2011)
Acknowledgements
The second author is very much indebted to Jorge Aragona for introducing him into the subject of Colombeau generalized functions in 1997 and for his collaboration throughout the years following this. His ideas and mathematical precision are the things which will be remembered. He is also indebted to J.F. Colombeau for many valuable teachings and collaborations. It was a privilege to have learned from the creator and masters of the theory.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Claudio Gorodski.
In honor of Alfredo Jorge Aragona Vallejo (In memoriam).
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Garcia, A.R.G., Juriaans, S.O., Oliveira, J. et al. A nonlinear theory of generalized functions. São Paulo J. Math. Sci. 16, 396–405 (2022). https://doi.org/10.1007/s40863-020-00205-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40863-020-00205-0