First lecture

On October 3, we tried to answer the general question What is a PDE?, using the following examples:

• The Cauchy-Riemann equations $u_x=v_y, \quad v_x=-u_y$ satisfied by the real an imaginary parts $u,v$ of an analytic function $f=u+iv$ of a complex variable;
• The wave equation (which, you may note, has solutions of class $C^2$ that are nowhere three times differentiable);
• The heat equation $u_t=\Delta u$; its derivation from two simple physical postulates and solutions of an initial-boundary value problem in 1 spatial dimension via Fourier series; smoothness of solutions (via term by term differentiation of the series, which is possible under mild assumptions);
• The minimal graph equation;
• The Laplace equation $\Delta u = 0$.

We also started looking at harmonic functions in more detail and discussed:

• the mean value property (for harmonic and subharmonic functions) and its proof;
• the maximum principle and its corollary: uniqueness for the Dirichlet problem.