We study the first Dirichlet eigenfunction of the Laplacian in a $n$-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti imply that the ratio of the $L^2$-norm and $L^{\infty}$-norm of the eigenfunction is minimized when the domain is a ball. However, when the eccentricity of the domain is large the eigenfunction should spread out at a certain scale and this ratio should increase. We make this precise by obtaining a lower bound on the $L^2$-norm of the eigenfunction and show that the eigenfunction cannot localize to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, in the general $n$-dimensional case. The main feature of the proof is to obtain sufficiently sharp estimates on the first eigenvalue in order to estimate the first derivatives of the eigenfunction.

中文翻译：

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凸域第一特征函数的定位

我们在 $n$ 维凸域中研究拉普拉斯算子的第一个 Dirichlet 特征函数。对于固定内半径的域，Chiti 的估计意味着当域是一个球时，本征函数的 $L^2$-范数和 $L^{\infty}$-范数的比率被最小化。然而，当域的偏心率很大时，特征函数应该在一定的尺度上展开，这个比例应该增加。我们通过获得本征函数的 $L^2$-范数的下界来使这一点精确，并表明本征函数不能定位到域的太小的子集。因此，我们在一般 $n$ 维情况下解决了 van den Berg 的猜想。证明的主要特征是对第一特征值获得足够清晰的估计，以便估计特征函数的一阶导数。