## 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].

### 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
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