关于Elliptic Differential Equations正解的边界表现
On the Boundary Behavior of Positive Solutions of Elliptic Differential Equations
摘要：
设$u$是单位球$B_1 \subset \mathbb{R}^n$中的正调和函数，$μ$是$u$的边界测度.考虑一个点$x\in \partial B_1$，并让$n(x)$表示$x$处的单位法向量。令$α$是$(-1,n-1]$和$A \in [0,+\infty) $中的一个数。证明了$u(x+n(x)t)t^α \to A$为$t \to +0$当且仅当$\frac{μ({B_r(x)})}{r^{n-1}} r^α \to C_αA$为$r\to+0$，其中${C_α= \frac{π^{n/2}}{Γ(\frac{n-α+1}{2})Γ(\frac{α+1}{2})}}$ .对于$α=0$它遵循的定理鲁丁和卢米斯声称，正调和函数有一个限制沿正常的当且仅当边界措施的衍生物在相应的点的边界。对于$α=n-1$，它涉及$x$中$μ$的点质量，并且它遵循Beurling极小原理。对于一般情况下的$α\in (-1,n-1)$，我们证明它的帮助下，维纳Tauberian定理以类似的方式鲁丁的方法。不幸的是，这种方法只适用于球或半空间，而不适用于一般类型的域。在维度$2$可以使用保形映射和推广上述声明充分光滑的域，在维度$n\geq 3$，我们表明，这种推广是可能的$α\in [0,n-1]$由于调和措施估计。最后一种方法将Loomis，Ramey和Ullrich关于调和函数的非切向极限的定理推广到具有Holder连续系数的椭圆型微分方程的正解.
英文摘要：
Let $u$ be a positive harmonic function in the unit ball $B_1 \subset \mathbb{R}^n$ and let $μ$ be the boundary measure of $u$. Consider a point $x\in \partial B_1$ and let $n(x)$ denote the unit normal vector at $x$. Let $α$ be a number in $(-1,n-1]$ and $A \in [0,+\infty) $. We prove that $u(x+n(x)t)t^α \to A$ as $t \to +0$ if and only if $\frac{μ({B_r(x)})}{r^{n-1}} r^α \to C_αA$ as $r\to+0$, where ${C_α= \frac{π^{n/2}}{Γ(\frac{n-α+1}{2})Γ(\frac{α+1}{2})}}$. For $α=0$ it follows from the theorems by Rudin and Loomis which claim that a positive harmonic function has a limit along the normal iff the boundary measure has the derivative at the corresponding point of the boundary. For $α=n-1$ it concerns about the point mass of $μ$ at $x$ and it follows from the Beurling minimal principle. For the general case of $α\in (-1,n-1)$ we prove it with the help of the Wiener Tauberian theorem in a similar way to Rudin's approach. Unfortunately this approach works for a ball or a half-space only but not for a general kind of domain. In dimension $2$ one can use conformal mappings and generalise the statement above to sufficiently smooth domains, in dimension $n\geq 3$ we showed that this generalisation is possible for $α\in [0,n-1]$ due to harmonic measure estimates. The last method leads to an extension of the theorems by Loomis, Ramey and Ullrich on non-tangential limits of harmonic functions to positive solutions of elliptic differential equations with Holder continuous coefficients.
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