Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

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.