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Mathematics LibreTexts

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), xΩRn, 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


This page titled 7: Elliptic Equations of Second Order is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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