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Some Phragmén-Lindelöf type results for the biharmonic equation

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Abstract

We derive estimates for weighted and unweighted energy expressions for the solution of a biharmonic boundary value problem in the plane by means of a second order differential inequality. We consider three kinds of unbounded domains with boundary conditions of Dirichlet type. For each domain we develop exponential estimates that either grow or decay. In the case of decay we also present a method for obtaining explicit bounds for the total energy.

Zusammenfassung

Wir leiten Abschätzungen für gewichtete und ungewichtete Energieausdrücke für die Lösung einer biharmonischen Randwertaufgabe in der Ebene mittels einer zweite Ordnung Differentialungleichung ab. Wir betrachten drei Arten von unbeschränkten Gebieten mit Randbedingungen des Dirichletschen Typs. Für jedes Gebiet entwickeln wir exponential Abschätzungen mit entweder positivem oder negativem Wachstum. Im Falle des negativem Wachstums geben wir auch eine Methode an, um explizite Abschätzungen für die totale Energie zu erhalten.

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Authors and Affiliations

  1. Dept of Mathematics, Cornell University, 14853, Ithaca, NY

    L. E. Payne

  2. Dept of Mathematics, University of Tennessee, 37996, Knoxville, Tennessee, USA

    P. W. Schaefer

Authors
  1. L. E. Payne
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  2. P. W. Schaefer
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Additional information

L. E. Payne was supported in part by the Tennessee Science Alliance while the author held a visiting position at the University of Tennessee.

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Payne, L.E., Schaefer, P.W. Some Phragmén-Lindelöf type results for the biharmonic equation. Z. angew. Math. Phys. 45, 414–432 (1994). https://doi.org/10.1007/BF00945929

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  • DOI: https://doi.org/10.1007/BF00945929

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