Dense Subsets of L1-Solutions to Linear Elliptic Partial Differential Equations

✍ Scribed by Henrik Shahgholian

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 273 KB
• Volume: 102
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1999.3408

✦ Synopsis

Let 0/R N (N 2) be an unbounded domain, and L m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L 1 -solutions to L m (in 0) approximate all L 1 -solutions to L m (in 0), provided there exist real numbers R j Ä , = 0, and a sequence [ y j ] such that B( y j , =) & 0=< and

where | } | means the volume and 4(z, R, D) := .

x

for z # R N , R>0 and D/R N . For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to L m may vanish, along with its (m&1)derivatives. We also study a similar approximation problem for polyanalytic functions in C.