Analysis and PDE : Harmonic function

$\Delta u = 0$ may be one of the most important function in the PDE since it appears many times in different context and has nice properties : well, we have to say that its beautiful property implies the interesting result in physics and our natural. Here, we recall some basic property and proof strategy of this topic.

Classical solution

Although the existence should be considered as the first property necessary for study the object, maybe it's nicer to give some interesting property at first. We suppose that the solution is of class $C^2$, then one important formula is Stokes formula that 

$$\begin{eqnarray*}\int_{\partial \Omega} F \cdot \nu  d\sigma = \int_{\Omega} \nabla \cdot F(x) dx \end{eqnarray*}$$

$$\int_{\partial \Omega} u \nu  d\sigma = \int_{\Omega} \nabla u(x) dx$$

we apply this and obtain that in a domain of harmonic function we have 

$$\int_{\partial \Omega} \nabla u(x) \cdot \nu d\sigma = \int_{\Omega} \Delta u(x) dx = 0$$

Furthermore, we obtain some useful formula as Green formula, we obtain the mean value principle that 

$$u(x) = \frac{1}{|\partial B_R(x)|}\int_{\partial B_R(x) }u(y) dy$$

One remarkable result is that this property improve the regularity as $C^{\infty}$ and it also implies the function is harmonic. (The proof is similar by the convolution below). Since if we do derivative, we get 

$$0 = \frac{d}{dr} \int_{\partial B_1(x) }u(x + ry) d\sigma = \int_{\partial B_1(x) }\nabla u(x + ry) \cdot \nu d\sigma = \frac{d}{dr} \int_{ B_r(x) } \Delta u(y) dy$$

A second important property is the Liouville proeprty. It says that a bounded harmonic function is trivial and is constant. The proof uses the fact that all the derivative are also harmonic 

$$|\partial_i u| \leq \frac{1}{\omega_d R^d} \int_{\partial B_R} |u| d\sigma \leq \frac{N}{R}\sup|u| \rightarrow 0$$

and then we use the mean value principle to analysis its size.

A third important property is the maximum principle. Idea is simple : the maximum and minimum of the function can only be attended at its boundary. Use the maximum principle, we prove that the uniqueness of the Dirichlet problem.

Weak solution is also classical solution

One very famous theorem Weyl states that all the weak solution that 

$$\int_{\Omega} u(x) \Delta \phi(x) dx = 0$$

for the test function $\phi \in C_c^{\infty}(\Omega)$ is also a strong solution of class $C^{\infty}$. The proof is very classical : we do convolution $u_{\epsilon} = u \ast \psi_{\epsilon}$. Then we prove that this function satisfies the harmonic by weak relation. Finally,  we pass this limit to the mean value principle. 

$$u_{\epsilon}(x) = \frac{1}{| B_R(x)|}\int_{ B_R(x) }u_{\epsilon}(y) dy \rightarrow u(x) = \frac{1}{| B_R(x)|}\int_{ B_R(x) }u(y) dy$$

Since this property doesn't require the regularity, we reprove that is is strong solution.

Existence

Finally, we come back to the problem of existence. There are two ways to prove the existence and it works in some more general framework. 

First one is the Lax-Milgram theorem. It treat the problem as find the inverse of operator in some function space 

$$a(u, v) = L(v)$$

The second one is the variational formation that we treat the solution  as a minimum of 

$$J(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^2 dx - \int_{\Omega} f u dx$$

One can also deduce the characterization from one to another.