Generalized solutions in PDEs and the Burgers' equation

Benci, Vieri; Baglini, Lorenzo Luperi

doi:10.1016/j.jde.2017.07.034

Cited by 9 publications

(4 citation statements)

References 26 publications

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“…Moreover, we will state a generalization of Gauss' divergence theorem which can be applied to the study of partial differential equations (see e.g. [9]). Here we give a simple application to equation (1) using an elementary notion of generalized solution (see section 4.3).…”

Section: Introductionmentioning

confidence: 99%

A generalization of Gauss’ divergence theorem

Benci¹,

Baglini²

2016

Recent Advances in Partial Differential Equations and Applications

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This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently in [2] and developed in [4], [5] and [6]. Their peculiarity is that they are based on a non-Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions.

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Section: Introductionmentioning

confidence: 99%

A generalization of Gauss’ divergence theorem

Benci¹,

Baglini²

2016

Recent Advances in Partial Differential Equations and Applications

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“…In this paper we deal with ultrafunctions, which are a kind of generalized functions that have been introduced recently in [1] and developed in [2,[4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Generalized solutions of variational problems and applications

Benci¹,

Baglini

Squassina

2018

Advances in Nonlinear Analysis

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Ultrafunctions are a particular class of generalized functions defined on a hyperreal field R * ⊃ R that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.2010 Mathematics Subject Classification. 03H05, 26E35, 28E05, 46S20.

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“…In a previous series of papers ([5], [10], [11], [12], [13], [14], [15]) we have introduced and studied a new family of generalized functions called ultrafunctions and its applications to certain problems in mathematical analysis, including some applications to PDE's in [15]. The development of a rigorous study of (a large class of) PDE's in ultrafunction theory is the object of [16], where we exemplify our approach by studying in detail Burgers' equation. Henceforth, it is our feeling that many problems in PDE's theory could be fruitfully studied by means of the theory of ultrafunctions.…”

Section: Introductionmentioning

confidence: 99%

A Topological Approach to Non-Archimedean Mathematics

Benci

Baglini

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [22]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.

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Generalized solutions in PDEs and the Burgers' equation

Cited by 9 publications

References 26 publications

A generalization of Gauss’ divergence theorem

A generalization of Gauss’ divergence theorem

Generalized solutions of variational problems and applications

A Topological Approach to Non-Archimedean Mathematics

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