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The Rellich Inequality

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)

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.

Keywords

Convex Domain Valid Inequality Magnetic Potential Hardy Inequality Limit Circle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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