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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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