Applications to Partial Differential Equations

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 6 in Integral Geometry and Convolution Equations, 2003, pp 408-415 from Springer

Abstract: Abstract Throughout in this chapter we assume that n ⩾ 2. Let f ∊ C 2(ℝ n ), and let $$ u\left( {x,r} \right) = \mathcal{R}f\left( {x,r} \right),r > 0 $$ (see (1.1)). Formula (1.2) implies the following identity which is called the Darboux equation: 6.1 $$ \frac{{\partial ^2 u}} {{\partial r^2 }} + \frac{{n - 1}} {r}\frac{{\partial u}} {{\partial r}} = \Delta _x u. $$ The left hand side is just the radial part of the Laplasian in ℝ n . To prove (6.1) we introduce the function $$ F\left( {x,y} \right) = \mathcal{R}f\left( {x,\left| y \right|} \right),y \in \mathbb{R}^n $$ . Using (1.2), we obtain $$ \Delta _y F = \int\limits_{SO\left( n \right)} {\Delta _y f\left( {x + \tau y} \right)d\tau } = \int\limits_{SO\left( n \right)} {\Delta _x f\left( {x + \tau y} \right)d\tau } = \Delta _x F. $$ Then the equation (6.1) follows, since F is radial in the y-variable.

Keywords: Partial Differential Equation; Cauchy Problem; Heat Equation; Uniqueness Theorem; Free Boundary Problem (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_32

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