Vieri Benci. Ultrafunctions and generalized solutions

Natural Sciences / Mathematics / Differential equations

Submitted on: Jun 30, 2012, 03:32:02

Description: The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of

--u=u^{p-1} , u>0, p≥((2N)/(N-2))

with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics (NAM) and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.

The abstract of this article has been published in the "Intellectual Archive Bulletin" , June 2012, ISSN 1929-1329.

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