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Some observations on the first eigenvalue of the \(p\)-Laplacian and its connections with asymmetry. (English) Zbl 0991.35032
Let \(\Omega\subset \mathbb{R}^N\) be a bounded domain. For \(p\in(1, \infty)\) let \(\lambda_1= \lambda_1(p,\Omega) =\inf{\int_\Omega|Du|^p dx\over \int_\Omega |u|^p dx}\), where the infimum is taken over all \(u\in W_0^{1,p} (\Omega)\), \(u\neq 0\). This paper is devoted to the lower bound \(\lambda_1 (\Omega)\) on bounded domains in \(\mathbb{R}^2\). Let \(\lambda_1^*\) be the first eigenvalue for the ball of the same volume. Then the author shows that \(\lambda_1\geq \lambda_1^*(1+C\alpha (\Omega)^3)\) for some constant \(C\), where \(\alpha\) is the asymmetry of the domain. Hence the author obtains a lower bound sharper than previous results in this direction.

MSC:
35J60 Nonlinear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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