Torsion and ground state maxima: close but not the same

B. Benson, R. Laugesen, M. Minion, B. Siudeja
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引用次数: 6

Abstract

Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture. We construct counterexamples on the half-disk, by working with the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has merit, since the maxima of the torsion function and eigenfunction are unexpectedly close together. It is an open problem to quantify this closeness in terms of the domain and the nonlinearity.
扭转和基态最大值:接近但不相同
一个半线性泊松方程在凸域上正解的最大值点的位置是否与非线性的形式无关?西玛和德里克为这个令人惊讶的猜想找到了确凿的证据。利用扭转函数和第一狄利克雷特征函数,构造了半盘上的反例。在等腰直角三角形上,这个猜想又失败了。然而,这个猜想有其优点,因为扭转函数和本征函数的最大值出乎意料地接近在一起。用定义域和非线性来量化这种接近性是一个开放的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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