The Wiener test for higher order elliptic equations. (English) Zbl 1018.35024

Maz’ya, Vladimir

Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. III: Invited lectures. Beijing: Higher Education Press. 189-195 (2002).

The author considers the following Dirichlet problem \[ L (\partial) u=f,\;f\in C^\infty_0 (\Omega),\;u\in {\overset\circ H}{^m} (\Omega), \tag{1} \] where \(L(\partial)\) is an elliptic differential operator of order \(2m\) in the Euclidean space \(\mathbb{R}^n\) with constant real coefficients, i.e., \[ L(\partial)= (-1)^m\sum_{|\alpha|= |\beta|=m} a_{\alpha \beta} \partial^{\alpha+ \beta}, \] here, \(\partial\) denotes the gradient \((\partial_{x_1},\dots, \partial_{x_n})\), \(a_{\alpha\beta}= a_{\beta\alpha}\), \((-1)^mL(\xi)>0\), \(\xi\in\mathbb{R}^n \setminus\{0\}\), \(\Omega\) is an open set in \(\mathbb{R}^n\), and \(\overset\circ H^m(\Omega)\) is the completion of \(C_0^\infty (\Omega)\) in the energy norm. He introduces the following definition of regularity (which is equivalent to that given by Wiener in the case \(m=1)\): A point \(x_0\in \partial\Omega\) is called regular with respect to \(L(\partial)\) if for any \(f\in C_0^\infty (\Omega)\) the solution of (1) satisfies \(\lim_{\Omega \ni x\to x_0}u(x)=0\), and states the following two theorems:
I: Let \(2m=n\). Then \(x_0\) is regular with resepct to \(L(\partial)\) if and only if \(\int_0^1 C_{2m} (B_\rho \setminus\Omega) \rho^{-1}d\rho= \infty\);
II: Let \(n>2m\) and let \(L(\partial)\) be positive with weight \(F\), that means \[ \int_{\mathbb{R}^n} L( \partial)u(x) \cdot u(x)F(x)dx\geq c\sum^m_{k=1} \int_{\mathbb{R}^n}\bigl|\nabla_k u(x)\bigr|^2|x|^{2k-m} dx \] for all real-valued \(u\in C_0^\infty(\mathbb{R}^n\setminus\{x_0\})\). Then \(x_0\) is regular with respect to \(L(\partial)\) if and only if \[ \int^1_0 C_{2m}(B_\rho\setminus\Omega)\rho^{2m-n-1} d\rho=\infty, \] here \(C_{2m}\) is the potential-theoretic Bessel capacity of order \(2m\). Additionally, several unsolved problems are presented at the end of the note.
For the entire collection see [Zbl 0993.00023].

MSC:

35J40	Boundary value problems for higher-order elliptic equations
35J30	Higher-order elliptic equations
31A15	Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31B15	Potentials and capacities, extremal length and related notions in higher dimensions

Keywords:

\(L\)-capacity potential; \(m\)-harmonic capacity; variational problem; Dirichlet problem; elliptic differential operator; definition of regularity; potential-theoretic Bessel capacity