Distributions and Fundamental Solutions

Norbert Ortner and Peter Wagner
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Norbert Ortner: University of Innsbruck, Department of Mathematics
Peter Wagner: University of Innsbruck, Faculty of Engineering Science

Chapter Chapter 1 in Fundamental Solutions of Linear Partial Differential Operators, 2015, pp 1-117 from Springer

Abstract: Abstract This chapter is an introduction to distribution theory illustrated by the verification of fundamental solutions of the classical operators $$\Delta _{n}^{k},(\lambda -\Delta _{n})^{k},(\Delta _{n}+\lambda )^{k},\partial _{\bar{z}}, (\partial _{t}^{2} - \Delta _{n})^{k},\partial _{1}\cdots \partial _{k},(\partial _{t} -\lambda \Delta _{n})^{k},(\partial _{t} \pm \text{i}\lambda \Delta _{n})^{k}$$ (for $$\lambda > 0,k \in \mathbf{N}),$$ which are listed in Laurent Schwartz’ famous book “Théorie des distributions,” see Schwartz [246]. The theory of distributions was developed by L. Schwartz in order to provide a suitable tool for solving problems in the analysis of several variables, e.g., in the theory of partial differential equations or in many-dimensional harmonic analysis. Taking into account this emphasis on many dimensions, we present mainly examples in R n instead of R 1. For example, in Example 1.4.10 we derive the distributional differentiation formula $$\displaystyle{\partial _{j}\partial _{k}{\Bigl (\frac{\vert x\vert ^{2-n} - 1} {2 - n} \Bigr )} =\mathop{ \mathrm{vp}}\nolimits {\Bigl ( \frac{\vert x\vert ^{2}\delta _{jk} - nx_{j}x_{k}} {\vert x\vert ^{n+2}} \Bigr )} + \frac{\vert \mathbf{S}^{n-1}\vert } {n} \,\delta _{jk}\delta,\quad 1 \leq j,k \leq n,\ n\neq 2.}$$

Date: 2015
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DOI: 10.1007/978-3-319-20140-5_1

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