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An inequality is proved for a quadratic functional with the logarithmic kernel. The best constant of this inequality and the corresponding function for which the equality holds are found precisely. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 3 &4 (2009), pp. 155 – 158 AN INEQUALITY FOR A QUADRATIC FUNCTIONAL K.C. Le Lehrstuhl f¨ ur Allgemeine Mechanik, Ruhr-Universit¨ at Bochum, 44780 Bochum, Germany Abstract. An inequality is proved for a quadratic functional with the logarithmic kernel. The best constant of this inequality and the corresponding function for which the equality holds are found precisely. The aim of this short communication is to provide one inequality for the following quadratic functional Z bZ b − ln |x − y| ϕ0 (x)ϕ0(y) dxdy, (1) a a where ϕ(x) are functions defined on the finite interval (a, b) such that p ϕ(x) = φ(x)w(x), w(x) = (b − x)(x − a), ˜ (a, b) be the space of all such functions. The norm in this with φ(x) ∈ W 1,2 (a, b). Let W function space is defined as the weighted norm kϕkW ˜ = kϕ/wkW 1,2 = kφkW 1,2 . The quadratic functional (1) appears in connection with various problems of physics and mechanics, for instance, the crack problems [1, 2, 3, 4, 5, 6], the dislocation pile-up problems [7, 8, 9], and the Peierls-Nabarro and Benjamin-Ono equations (see [10, 11] and the references therein). Because of the singularity of the logarithmic kernel, the double integral in (1) should be defined as Z bZ b Z 0 0 − ln |x − y| ϕ (x)ϕ (y) dxdy = − lim ln |x − y| ϕ0(x)ϕ0 (y) dxdy, (2) a a ε→0 Sε where Sε is the square (a, b) × (a, b) in the (x, y)-plane with the diagonal band of height 2ε being removed Sε = {(x, y)| a ε}. Note that, by the partial integration with respect to x, one can present (2) in the form Z bZ b Z bZ b 0 ϕ (y) − ln |x − y| ϕ0(x)ϕ0 (y) dxdy = − dyϕ(x) dx, (3) − y −x a a a a 156 K.C. Le R where − denotes Cauchy’s principal value of the integral. Note also the close connection of (3) with the capacity of the logarithmic potential induced by a 2-D continuous charge distribution on the interval (a, b). It turns out that the following inequality holds true for this functional: there exists ˜ (a, b) a positive
