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

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].


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
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