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On an Inequality of C. Sundberg: A Computational Investigation via Nonlinear Programming

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Abstract

The main goal of this article is to discuss a numerical method for finding the best constant in a Sobolev type inequality considered by C. Sundberg, and originating from Operator Theory. To simplify the investigation, we reduce the original problem to a parameterized family of simpler problems, which are constrained optimization problems from Calculus of Variations. To decouple the various differential operators and nonlinearities occurring in these constrained optimization problems, we introduce an appropriate augmented Lagrangian functional, whose saddle-points provide the solutions we are looking for. To compute these saddle-points, we use an Uzawa–Douglas–Rachford algorithm, which, combined with a finite difference approximation, leads to numerical results suggesting that the best constant is about five times smaller than the constant provided by an analytical investigation.

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Acknowledgements

This work was supported by the NSF (grant DMS-0913982).

The authors would like to thank Professor C. Sundberg, UT-Knoxville, for suggesting them to look at problem (1). They also thank X. Feng and S. Poole for the organization of the 2010 visit of the first author at ORNL and UT-Knoxville, Professor F. Giannessi and the two anonymous referees for helpful comments and suggestions. The support of University of Tennessee—Knoxville, ORNL and NSF (grant DMS-0913982) is also acknowledged.

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  1. Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, 77204, TX, USA

    R. Glowinski & A. Quaini

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  1. R. Glowinski
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  2. A. Quaini
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Correspondence to A. Quaini.

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Dedicated to Richard Tapia.

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Glowinski, R., Quaini, A. On an Inequality of C. Sundberg: A Computational Investigation via Nonlinear Programming. J Optim Theory Appl 158, 739–772 (2013). https://doi.org/10.1007/s10957-013-0275-y

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