Remark 12.1.2 (Examples).label

• $\la u=\sums{n }{i=1 }\pd_{i}^{2}u=b$ is the Poisson equation or Laplace equation for $b=0$ (Laplacian)

• $\square u=\pd_{1}^{2}u-\sums{n }{j=2}\pd_{j}^{2}u=0$ Wave equation (D’Alembertion)

• $\pd_{1}u=\sums{n }{j=2}\pd_{j}^{2}u=0$ Heat equation

We note that these equations behave very differently. Fix $n=2$ then

• $\la u=0,u\in C^{2}\implies u\in C^{\om}$. If $u$ is defined on $\R^{2}$ and $\SUP{x_1,x_2\in \R^2}\abs{u(x_1,x_2)}<\infty$ then $u$ is constant.

• $\square u=0,u=f(x_{1}+x_{2})+g(x_{1}-x_{2})$ viewing $x_{2}$ as a time variable and $u$ is discontinuous at some point $(x_{1}^{*},x_{2}^{*})$ then discontinuity propagates

• If initial condition $u(x_{1},0)=h(x_{1})$ is $C^{0}$ with $u$ solving heat equation then $u(x_{1},x_{2})\in C^{\infty}$