1 results
A Lebesgue–Lusin property for linear operators of first and second order
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 06 November 2023, pp. 1-11
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- Article
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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.
