# Boundary Value Problems

The fundamental solution to the **Poisson equation** with respect to a point-like excitation is known. Such a solution is called **Green's function** - in three dimensions, we have the famous \(1/r\)-dependency. However, if one knows the Green's function for a differential operator (here Laplace \(\Delta\)) any problem can be solved with respect to the given **boundary conditions**. In this section we will learn different techniques to solve such boundary value problems, for example the method of mirror charges or direct integrations.

The Green's function can be used to find the electric field or electrostatic potential for arbitrary **charge distributions**. You will learn that the **polarization** of a **dipole** in front of a flat metallic surface enormously affects the overall field.

The so-called **boundary integral equation** relates the values of the electrostatic potential in some domain to its values at that domain's boundary. In this problem we will **derive** this important statement which leads to the "**Boundary Element Method**", a discretized version with **numerical applications** throughout science and engineering.

In this problem we will encounter the main physical features of dielectric spheres - their **induced field** and **polarizability**. The result is of great interest: understanding the interaction of light with **small particles** is one of the main concerns of **nanophotonics**. It leads to astonishing applications like **cloaking**, improvement of **solar cells** and **lithography** with extreme resolution.

The **method of image charges** is a powerful technique to find the **electrostatic potential** in an intuitive way. In this problem you will find out how it can be applied to a point charge close to two grounded intersecting metallic half-planes.