Elliptic regularity with continuous and branching edge asymptotics.

*(English)*Zbl 0718.35012
Multivariate approximation and interpolation, Proc. Int. Workshop, Duisburg/FRG 1989, ISNM Int. Ser. Numer. Math. 94, 275-284 (1990).

[For the entire collection see Zbl 0703.00022.]

The paper gives a description of the asymptotics of solutions of elliptic equations in non-smooth domains where the singular set is of the type of an edge, outside of which the structure is \(C^{\infty}\). The difficulty is that the data of the asymptotics \[ \sum^{\infty}_{j=0}\sum^{m_ j}_{k=0}\zeta_{jk}r^{-p_ j} \log^ kr\quad as\quad r\to 0 \] with r being the ‘distance’ to the edge, such as the \(p_ j\in {\mathbb{C}}\), \(m_ j\in {\mathbb{N}}\), and the coefficients \(\zeta_{jk}\in {\mathbb{C}}\) may depend on the edge variable y, and there are in general jumps or more chaotic changes with varying y. This is to be expected even for solutions which are \(C^{\infty}\) outside the edge. In the author’s work [Integral Equations Oper. Theory 11, No.4, 557-602 (1988; Zbl 0671.58040)] the singular functions have been described in this \(C^{\infty}\) case, by using \(C^{\infty}\) functions of analytic functionals in the complex Mellin plane which are pointwise discrete and of finite order. The present article extends that result to solutions in weighted Sobolev spaces. Background are the more general continuous asymptotics, earlier introduced by the author [Math. Nachr. 136, 7-57 (1988; Zbl 0664.58041)].

The paper gives a description of the asymptotics of solutions of elliptic equations in non-smooth domains where the singular set is of the type of an edge, outside of which the structure is \(C^{\infty}\). The difficulty is that the data of the asymptotics \[ \sum^{\infty}_{j=0}\sum^{m_ j}_{k=0}\zeta_{jk}r^{-p_ j} \log^ kr\quad as\quad r\to 0 \] with r being the ‘distance’ to the edge, such as the \(p_ j\in {\mathbb{C}}\), \(m_ j\in {\mathbb{N}}\), and the coefficients \(\zeta_{jk}\in {\mathbb{C}}\) may depend on the edge variable y, and there are in general jumps or more chaotic changes with varying y. This is to be expected even for solutions which are \(C^{\infty}\) outside the edge. In the author’s work [Integral Equations Oper. Theory 11, No.4, 557-602 (1988; Zbl 0671.58040)] the singular functions have been described in this \(C^{\infty}\) case, by using \(C^{\infty}\) functions of analytic functionals in the complex Mellin plane which are pointwise discrete and of finite order. The present article extends that result to solutions in weighted Sobolev spaces. Background are the more general continuous asymptotics, earlier introduced by the author [Math. Nachr. 136, 7-57 (1988; Zbl 0664.58041)].

Reviewer: B.-W.Schulze