
7: Elliptic Equations of Second Order


Here we consider linear elliptic equations of second order, mainly the Laplace equation

$$\triangle u=0. \] Solutions of the Laplace equation are called potential functions or harmonic functions. The Laplace equation is called also potential equation. The general elliptic equation for a scalar function $$u(x)$$, $$x\in\Omega\subset\mathbb{R}^n$$, is$$Lu:=\sum_{i,j=1}^na^{ij}(x)u_{x_ix_j}+\sum_{j=1}^n b^j(x)u_{x_j}+c(x)u=f(x),\]

where the matrix $$A=(a^{ij})$$ is real, symmetric and positive definite. If $$A$$ is a constant matrix, then a transform to principal axis and stretching of axis leads to

\sum_{i,j=1}^na^{ij}u_{x_ix_j}=\triangle v,\]

where $$v(y):=u(Ty)$$, $$T$$ stands for the above composition of mappings.