7: Elliptic Equations of Second Order

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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.

Contributors and Attributions

Prof. Dr. Erich Miersemann (Universität Leipzig)
Integrated by Justin Marshall.