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.

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