Opuscula Mathematica

Opuscula Math.
 28
, no. 4
 (), 561-566
Opuscula Mathematica

Some remarks on the optimization of eigenvalue problems involving the p-Laplacian


Abstract. Given a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u \) with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration \(D\) is connected with the corresponding positive eigenfunction \(u\) in such a way that there exists a number \(t\geq 1\) for which \(D=\{u \leq t\}\). We also give a new proof of symmetry of optimal solutions in the case when \(\Omega \) is Steiner symmetric and \(p = 2\).
Keywords: \(p\)-Laplacian, the first eigenvalue, Steiner symmetry.
Mathematics Subject Classification: 35P30, 35J65, 35J70.
Cite this article as:
Wacław Pielichowski, Some remarks on the optimization of eigenvalue problems involving the p-Laplacian, Opuscula Math. 28, no. 4 (2008), 561-566
 
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.