Every positive regular solution of the integral equation posed by Lieb is radially symmetric and monotone

Analysis 1 Minute

You need to know: Euclidean space {\mathbb R}^n, integration over {\mathbb R}^n, space of functions L^p({\mathbb R}^n) (functions u:{\mathbb R}^n\to{\mathbb R} such that \int_{{\mathbb R}^n}|u|^p < \infty).

Background: For positive integer n and 0 < \alpha < n, consider the integral equation u(x) = \int_{{\mathbb R}^n}\frac{1}{|x-y|^{n-\alpha}}u(y)^{\frac{n+\alpha}{n-\alpha}}dy. We call its solution u regular if u\in L^{\frac{2n}{n-\alpha}}({\mathbb R}^n).

The Theorem: In August 2004, Wenxiong Chen, Congming Li, and Biao Ou submitted to the Communications on Pure and Applied Mathematics a paper in which they proved that every positive regular solution of the integral equation above has the form u(x) = c\left(\frac{t}{t^2+|x-x_0|^2}\right)^{\frac{n-\alpha}{2}}, with some constant c = c(n, \alpha) and some t > 0 and x_0 \in {\mathbb R}^n.

Short context: Integral equation above arose in 1983 paper of Lieb on best possible constant in so-called Hardy-Littlewood-Sobolev inequality. It also has connection with a well-known family of semilinear partial differential equations. Lieb posed the classification of all the solutions of this integral equation as an open problem. This problem was open for over 20 years, until was fully solved by the Theorem.

Links: The original paper is available here.

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Published by Bogdan Grechuk

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