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Title of Journal: J Optim Theory Appl

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Abbravation: Journal of Optimization Theory and Applications

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Springer US

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DOI

10.1007/s002140000243

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1573-2878

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On an Inequality of C Sundberg A Computational I

Authors: R Glowinski A Quaini
Publish Date: 2013/02/05
Volume: 158, Issue: 3, Pages: 739-772
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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 saddlepoints provide the solutions we are looking for To compute these saddlepoints 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 investigationThe authors would like to thank Professor C Sundberg UTKnoxville 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 UTKnoxville Professor F Giannessi and the two anonymous referees for helpful comments and suggestions The support of University of Tennessee—Knoxville ORNL and NSF grant DMS0913982 is also acknowledged

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