Ultrafunctions and Generalized Solutions

作者：Benci Vieri^*

来源：Advanced Nonlinear Studies, 2013, 13(2): 461-486.

DOI：10.1515/ans-2013-0212

• Non Archimedean Mathematics
• Non Standard Analysis
• ultrafunctions
• distributions
• generalized solutions
• Sobolev critical exponent
• formalism of Quantum Mechanics

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 %26lt;br%26gt;-Delta u = u(p-1), u %26gt; 0, p %26gt;= 2N/N - 2 %26lt;br%26gt;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.

• 出版日期2013-5