Poisson 积分公式

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Schwarz and Poisson formulas
Differential Equations for Engineers - Fourier series and PDEs- Dirichlet problem in the circle and the Poisson kernel

$$u(r, \theta)= \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1-r^2}{1-2r\cos(\theta-\alpha)+r^2}g(\alpha)d\alpha.  $$
让我们解释一下泊松核的几何意义。极坐标中 $(r,θ)$ 到 $(1,α)$ 的距离 $s$ 是$1-2r\cos(θ-α)+r^2$ 的平方根。也就是说，泊松核实际上是 $\frac{1-r^2}{s^2}$。最后请注意，该公式实际上是边界值的加权平均。首先让我们看看在原点发生了什么，即 $r=0$ 时。\begin{aligned} u(0,0) &= \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1-0^2}{1-2(0)\cos(\theta-\alpha)+0^2}g(\alpha)d\alpha \\ &= \frac{1}{2\pi} \int_{-\pi}^{\pi} g(\alpha)d\alpha. \end{aligned}所以 $u(0,0)$ 正是 $g(θ)$ 的平均值，因此是边界上 $u$ 的平均值。 这是调和函数的一般性质，在某个点 $p$ 的值等于以 $p$ 为中心的圆周上的值的平均值。Gauss's Mean Value Theorem
Poisson 积分公式 表明，圆中任意点的解函数的值是边界数据 $g(α)$ 的加权平均值。当 $(r,θ)$ 更接近 $(1,α)$ 时，泊松核更大。 因此，在计算 $u(r,θ)$ 时，当 $(1,α)$ 更接近 $(r,θ)$ 时，赋予 $g(α)$ 更多权重；当 $(1,α)$ 更远离 $(r,θ)$ 时，赋予 $g(α)$ 更少的权重。

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Lesson 33, 34: The Interior and Exterior Dirichlet Problem for a Circle

• It is interesting to note that from this formula, it can easily be seen that
  $$
  u(0,0)=\frac{1}{2 \pi} \int_0^{2 \pi} g(\alpha) d \alpha .
  $$
• That is, the value of the solution at the center of the circle is equal to the average value of $g$ on the boundary.
• Furthermore, by the Law of Cosines, the denominator in the integrand is the square of the length of the side opposite the angle $|\theta-\alpha|$ of the triangle with vertices $(0,0),(r, \theta)$, and $(R, \alpha)$.
• For a BVP defined on a non-circular domain, this formula can be applied by first conformally mapping the domain to a circle, using the formula on the transformed problem, and then transforming back to the original domain.