The Rellich Inequality

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


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
$$\displaystyle{ \int _{\mathbb{R}^{n}}\vert \Delta u(\mathbf{x})\vert ^{2}d\mathbf{x} \geq \frac{n^{2}(n - 4)^{2}} {16} \int _{\mathbb{R}^{n}}\frac{\vert u(\mathbf{x})\vert ^{2}} {\vert \mathbf{x}\vert ^{4}} d\mathbf{x},\ \ u \in C_{0}^{\infty }(\mathbb{R}^{n}\setminus \{0\}), }$$
while for n = 2, the inequality continues to hold but for a restricted class of functions u; see Remark 6.4.4 below.


Convex Domain Valid Inequality Magnetic Potential Hardy Inequality Limit Circle 
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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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