The Rellich Inequality

Abstract

In lectures delivered at New York University in 1953, and published posthumously in the proceedings [128] of the International Congress of Mathematicians held in Amsterdam in 1954, Rellich proved the following inequality which bears his name: for n ≠ 2

$$\displaystyle{ \int _{\mathbb{R}^{n}}\vert \Delta u(\mathbf{x})\vert ^{2}d\mathbf{x} \geq \frac{n^{2}(n - 4)^{2}} {16} \int _{\mathbb{R}^{n}}\frac{\vert u(\mathbf{x})\vert ^{2}} {\vert \mathbf{x}\vert ^{4}} d\mathbf{x},\ \ u \in C_{0}^{\infty }(\mathbb{R}^{n}\setminus \{0\}), }$$

(6.1.1)

while for n = 2, the inequality continues to hold but for a restricted class of functions u; see Remark 6.4.4 below.