论文标题
$ s> 1/2 $的球上不在$ h^s $中的连续谐波功能
Continuous harmonic functions on a ball that are not in $H^s$ for $s>1/2$
论文作者
论文摘要
我们表明，$ {\ mathbb {\ mathbb {b} _n} $ of $ \ mathbb {r}^n $，$ n \ ge 2 $的谐波功能是连续到边界的（甚至连Hölder连续）的连续（甚至在sobolev space $ h^s $ h^s（\ mathbbbb）中，这些功能构建的想法是受哈玛德（Hadamard）在1906年提供的无限能量的谐波连续功能的二维示例的启发。要在任何维度中获得示例$ n \ ge 2 $，我们利用某些系列的球形和声。作为一种应用，我们验证在某种意义上，对于一类非线性传输条件的边界价值问题所证明的解决方案的规律性是最佳的。
We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even Hölder continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$ sufficiently big. The idea for the construction of these functions is inspired by the two-dimensional example of a harmonic continuous function with infinite energy presented by Hadamard in 1906. To obtain examples in any dimension $n\ge 2$ we exploit certain series of spherical harmonics. As an application, we verify that the regularity of the solutions that was proven for a class of boundary value problems with nonlinear transmission conditions is, in a sense, optimal.
