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Kato’s Inequality in the Half Space: An Alternative Proof and Relative Improvements

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Geometric Properties for Parabolic and Elliptic PDE's

Part of the book series: Springer INdAM Series ((SINDAMS,volume 2))

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Abstract

In this paper we give an alternative proof of the optimal Kato’s inequality in the half space. The approach is based on a very classical method of Calculus of Variation due to Weirstrass (and developed by Hilbert) that usually is considered to prove that the solutions of the Euler Lagrange equation associated to a functional are, in fact, extremals. In this paper we will show how this method is well suited also to functionals that have no extremals. Moreover, we will present a class of inequalities that interpolate Kato’s inequality and Hardy’s inequality in the half space.

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Authors and Affiliations

  1. Dipartimento di Matematica, Seconda Università di Napoli, Viale Lincoln 5, 81100, Caserta, Italy

    Adele Ferone

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  1. Adele Ferone
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Correspondence to Adele Ferone .

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Editors and Affiliations

  1. Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, Firenze, 50124, Italy

    Rolando Magnanini

  2. Graduate School of Information Sciences, Division of Mathematics, Tohoku University, Sendai, 980-8579, Japan

    Shigeru Sakaguchi

  3. , Dipartimento di Matematica ed Applicazio, Università di Napoli Federico II, Via Cintia, Monte S. Angelo, Napoli, 80126, Italy

    Angelo Alvino

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Ferone, A. (2013). Kato’s Inequality in the Half Space: An Alternative Proof and Relative Improvements. In: Magnanini, R., Sakaguchi, S., Alvino, A. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2841-8_6

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