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Mathematics LibreTexts

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.