Looking for the best constant in a Sobolev inequality: a numerical approach

ISSN： 0008-0624

Source： CALCOLO, Vol.47, Iss.4, 2010-12, pp. : 211-238

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Abstract

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H ²(Ω) and $C^{0}(overline{Omega})$ is presented. Green's functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green's functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.

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