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# 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.

### Contributors

• Integrated by Justin Marshall.