Differential and Integral Equations

Applications of a one-dimensional Sobolev inequality to eigenvalue problems

R. C. Brown, D. B. Hinton, and Š. Schwabik

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A one-dimensional Sobolev-type inequality supplemented by a Prüfer transformation argument is used to derive upper and lower bounds for the eigenvalues of regular, self-adjoint second-order eigenvalue problems. These inequalities are shown to have applications to counting eigenvalues in the intervals $\scriptstyle (-\infty,\lambda]$, estimating eigenvalue gaps, Liapunov inequalities, and de La Valée Poussin-type inequalities.

Article information

Differential Integral Equations, Volume 9, Number 3 (1996), 481-498.

First available in Project Euclid: 7 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds
Secondary: 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)


Brown, R. C.; Hinton, D. B.; Schwabik, Š. Applications of a one-dimensional Sobolev inequality to eigenvalue problems. Differential Integral Equations 9 (1996), no. 3, 481--498. https://projecteuclid.org/euclid.die/1367969967

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