Verifying General Solution of 2D Poisson Equation

[glmuelle]

Hi

Homework Statement

Verify, that

$$u(\vec{x}) := - \frac{1}{2 \pi} \int \limits_{\mathbb{R}^2} \log ||\vec{x} - \vec{y} || f(\vec{y}) d \vec{y}$$

is the general solution of the 2 dimensional Poisson equation:

$$\Delta u = - f$$

where $$f \in C^2_c(\mathbb{R}^2)$$ is differentiable twice and has compact support.

Homework Equations

The Attempt at a Solution

My attempt would be to swap integral and Laplace operator but I know it's wrong to just do that...
Can anyone help me please? Thanks!
Gloria

 

[snapback]

the Laplace operator is applied with respect to x, and the integration is performed over y -> I think you can swap them ( x is only a parameter inside of the integral)