Graphs of Finite Mass Which Cannot Be Approximated by Smooth Graphs with Equibounded Area
✍ Scribed by Domenico Mucci
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 369 KB
- Volume
- 152
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
We show that there exist graphs of Cartesian maps, of finite mass, which cannot be approximated weakly as currents by graphs of smooth maps with equibounded area.
1998 Academic Press
This paper deals with the relaxed extension of the nonparametric area functional for vector valued maps.
Let 0 be a bounded domain in R n , n, N 2, and u: 0 Ä R N be a smooth map. The area of the graph of u over 0 is given by
where
is the square root of the sum of the squares of the determinants of all minors of the Jacobian matrix Du up to the order n Ä :=min(n, N). In the same spirit as Lebesgue's area for continuous functions, the relaxed area of the ``graph'' of an L 1 -function u: 0 Ä R N is defined by
Denote by A 1 (0, R N ) the class of maps u # W 1, 1 (0, R N ) such that all minors of Du are summable in 0. If u # A 1 (0, R N ), we can still define A(u, 0) by (1), so that due to the lower semicontinuity of A(u, 0) w.r. to the L 1 -convergence, see e.g. [1], one has A(u, 0) A (u, 0).