TY  - JOUR
ID  - Pielichowski2008
LB  - Pielichowski2008
AU  - Pielichowski, Wacław
TI  - Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
ST  - Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2008
VL  - 28
IS  - 4
SP  - 561
EP  - 566
DO  - http://dx.doi.org/10.7494/OpMath.2008.28.4.561
UR  - http://dx.doi.org/10.7494/OpMath.2008.28.4.561
AB  - 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 \(p = 2\).
KW  - \(p\)-Laplacian
KW  - the first eigenvalue
KW  - Steiner symmetry
SN  - 1232-9274
ER  -