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Generalized solutions of singular differential problems. Relationship with classical solutions. (English) Zbl 1169.35319
Summary: We discuss various methods of regularization of singular differential problems. Their common point is that we use the flexibility of the theories of nonlinear generalized functions for adapting the regularization to the singularity of the problem. We particularly underline the relationship between the generalized solutions and those classical or distribution, when they exist, giving a general result for the case of the regularization of data.

35G20 Nonlinear higher-order PDEs
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
35A20 Analyticity in context of PDEs
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[1] Antonevich, A.B.; Radyno, Ya.V., On a general method for constructing algebras of generalized functions, Soviet math. dokl., 43, 3, 680-684, (1992) · Zbl 0862.46019
[2] Biagioni, H.A., A nonlinear theory of generalized functions, Lecture notes in math., vol. 1421, (1990), Springer Berlin · Zbl 0807.35020
[3] Colombeau, J.-F., New generalized functions and multiplication of distributions, (1984), North-Holland Amsterdam
[4] Constantine, G.M.; Savits, H., A multivariate faà di bruno formula with applications, Trans. amer. math. soc., 348, 2, 503-520, (1996) · Zbl 0846.05003
[5] Delcroix, A., Remarks on the embedding of spaces of distributions into spaces of colombeau generalized functions, Novi sad J. math., 35, 2, 27-40, (2005) · Zbl 1274.46082
[6] Delcroix, A.; Marti, J.-A.; Oberguggenberger, M., Spectral asymptotic analysis in algebras of generalized functions, Asymptot. anal., 59, 1-2, 83-107, (2008) · Zbl 1163.35303
[7] Delcroix, A.; Scarpalézos, D., Topology on asymptotic algebras of generalized functions and applications, Monatsh. math., 129, 1-14, (2000) · Zbl 0952.46029
[8] Dévoué, V., On generalized solutions to the wave equation in canonical form, Dissertationes math., 443, 1-69, (2007) · Zbl 1195.35195
[9] Gelfand, I.; Shilov, G., Generalized functions, vols. 1 & 2, (1964 & 1968), Academic Press New York
[10] Grosser, M.; Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R., Geometric theory of generalized functions with applications to general relativity, (2001), Kluwer Academic Press Dordrecht · Zbl 0998.46015
[11] Hörmann, G.; De Hoop, M.-V., Microlocal analysis and global solutions for some hyperbolic equations with discontinuous coefficients, Acta appl. math., 67, 173-224, (2001) · Zbl 0998.46016
[12] S. Haller, G. Hörmann, Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients, Publ. Inst. Math. (Beograd) (N.S.), in press · Zbl 1274.35225
[13] Hörmander, L., The analysis of linear partial differential operators I: distribution theory and Fourier analysis, Grundlehren math. wiss., vol. 256, (1990), Springer Berlin · Zbl 0712.35001
[14] Hörmander, L., The analysis of linear partial differential operators III: pseudo-differential operators, Grundlehren math. wiss., vol. 256, (1990), Springer Berlin
[15] Marti, J.-A., \((\mathcal{C}, \mathcal{E}, \mathcal{P})\)-sheaf structure and applications, ()
[16] Marti, J.-A., Non-linear algebraic analysis of delta shock wave to Burgers’ equation, Pacific J. math., 210, 1, 165-187, (2003) · Zbl 1059.35122
[17] Marti, J.-A.; Nuiro, P.S.; Valmorin, V.S., A nonlinear Goursat problem with irregular data, Integral transforms spec. funct., 6, 1-4, 229-246, (1998) · Zbl 0912.35043
[18] Nedeljkov, M.; Pilipović, S.; Scarpalézos, D., The linear theory of colombeau generalized functions, Pitman res. notes math. ser., vol. 385, (1998), Longman Harlow · Zbl 0918.46036
[19] Oberguggenberger, M., Multiplication of distributions and applications to partial differential equations, (1992), Longman Scientific & Technical Harlow · Zbl 0818.46036
[20] Pilipović, S.; Scarpalézos, D., Divergent type quasilinear Dirichlet problem, Acta appl. math., 94, 67-82, (2006) · Zbl 1114.46030
[21] Scarpalézos, D., Colombeau’s generalized functions: topological structures; microlocal properties, A simplified point of view, part I, Bull. cl. sci. math. nat. sci. math., 25, 89-114, (2000) · Zbl 1011.46042
[22] Scarpalézos, D., Colombeau’s generalized functions: topological structures; microlocal properties, A simplified point of view, part II, Publ. inst. math. (beograd) (N.S.), 76, 90, 111-125, (2004) · Zbl 1221.46046
[23] Schwartz, L., Théorie des distributions, (1966), Hermann Paris
[24] Khoan, V.K., Distributions, analyse de Fourier, opérateurs aux Dérivées partielles, vol. 1, (1972), Vuibert Paris
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