论文标题
在不可压缩问题中的独特标准和对规律性的反例
A uniqueness criterion and a counterexample to regularity in an incompressible variational problem
论文作者
论文摘要
在本文中，我们考虑了形式$ e（u）= \ int_b f（x，x，\ nabla u）\，dx $在适当准备的一类不可压缩的，平面映射$ u：b \ rightArrow \ rightarrow \ mathbb {r}^2 $中的问题。在这里，$ b $是单位磁盘，$ f（x，ξ）$是二次磁盘，$ξ$ subenvex。结果表明，如果$ u $是纸张清楚地表明的$ e $的固定点，那么$ u $是$ e（u）$的独特全球最小化器，前提是相应压力的梯度满足适当的较小条件。我们将此结果应用于构建一个非自主的，均匀凸出的功能性$ f（x，ξ）$，具体取决于$ξ$，但在$ x $上不连续，其独特的全局最小化是所谓的$ n- $ covering Map，它是Lipschitz，但不是$ C^1 $。
In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,ξ)$ is quadratic and convex in $ξ$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,ξ)$, depending smoothly on $ξ$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.
