论文标题
第一和二阶线性运算符的Lebesgue-Lusin属性
A Lebesgue-Lusin property for linear operators of first and second order
论文作者
论文摘要
我们证明，对于均质的线性部分差分运算符$ \ MATHCAL $ k \ le 2 $和可集成的映射$ f $，在该操作员的基本范围内，存在一个特殊有限变化的函数$ u $ \ [ \ Mathcal a u（x）= f（x）\ qquad \ text {几乎无处不在}。 \]这扩展了G. alberti的结果，以$ \ mathbf r^n $上的梯度。特别是，对于$ 0 \ le m <n $，这表明每个可集成的$ m $ - 向量字段是普通$（m+1）$ - 当前边界的绝对连续部分。
We prove that for a homogeneous linear partial differential operator $\mathcal A$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying \[ \mathcal A u(x)= f(x) \qquad \text{almost everywhere}. \] This extends a result of G. Alberti for gradients on $\mathbf R^N$. In particular, for $0 \le m < N$, it is shown that every integrable $m$-vector field is the absolutely continuous part of the boundary of a normal $(m+1)$-current.
