A second order quadratic integral inequality associated with regular problems

Moumita Bhattacharyya; Shib Sankar Sana; Moumita Bhattacharyya; Shib Sankar Sana

doi:10.3934/mmc.2024013

Mathematical Modelling and Control

2024, Volume 4, Issue 1: 141-151. doi: 10.3934/mmc.2024013

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Research article

A second order quadratic integral inequality associated with regular problems

Moumita Bhattacharyya ¹,
Shib Sankar Sana ^{2
,
,}

1.
Department of Pure Mathematics, University of Calcutta, Ballygunge Circular Road 35, Kolkata 700019, India
2.
Department of Mathematics, Kishore Bharati Bhagini Nivedita College, Ramkrishna Sarani, Behala, Kolkata 700060, India

Received: 06 October 2023 Revised: 12 February 2024 Accepted: 18 February 2024 Published: 09 April 2024

In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all $ f\in D $,

$ \begin{eqnarray} \int_{a}^{b}{r|f''|^{2}+p|f'|^{2}+q|f|^{2}}\geq\mu_{0}\int_{a}^{b}|f|^{2}. \end{eqnarray} $

Here $ r, p, q $ are real- valued coefficient functions on the compact interval $ [a, b] $ with $ r(x) > 0 $. $ D $ is a linear manifold in the Hilbert function space $ L^{2}(a, b) $ such that all integrals of the above inequality are finite and $ \mu_{0} $ is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in $ L^{2}(a, b) $. The inequality is the best possible, i.e., the number $ \mu_{0} $ cannot be increased. $ f $ is a complex-valued function in $ D. $
- regular problems,
- self-adjoint differential operators
Citation: Moumita Bhattacharyya, Shib Sankar Sana. A second order quadratic integral inequality associated with regular problems[J]. Mathematical Modelling and Control, 2024, 4(1): 141-151. doi: 10.3934/mmc.2024013

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Abstract

In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all $ f\in D $,

$ \begin{eqnarray} \int_{a}^{b}{r|f''|^{2}+p|f'|^{2}+q|f|^{2}}\geq\mu_{0}\int_{a}^{b}|f|^{2}. \end{eqnarray} $

Here $ r, p, q $ are real- valued coefficient functions on the compact interval $ [a, b] $ with $ r(x) > 0 $. $ D $ is a linear manifold in the Hilbert function space $ L^{2}(a, b) $ such that all integrals of the above inequality are finite and $ \mu_{0} $ is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in $ L^{2}(a, b) $. The inequality is the best possible, i.e., the number $ \mu_{0} $ cannot be increased. $ f $ is a complex-valued function in $ D. $

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