7: Elliptic Equations of Second Order
( \newcommand{\kernel}{\mathrm{null}\,}\)
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∈Ω⊂Rn, 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=(aij) 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.
Contributors and Attributions
Integrated by Justin Marshall.