Elliptic partial differential equation

An elliptic partial differential equation is a general partial differential equation of second order of the form

Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,

that satisfies the condition

B^2 - AC < 0.\

(Assuming implicitly that u_{xy}=u_{yx}. )

Just as one classifies conic sections and quadratic forms based on the discriminant B^2 - 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B^2 - AC, due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

Ax^2 + 2Bxy + Cy^2 + \cdots = 0 , which becomes (for : u_{xy}=u_{yx}=0) :
Au_{xx} + Cu_{yy} + Du_x + Eu_y + F = 0 , and Ax^2 + Cy^2 + \cdots = 0 . This resembles the standard ellipse equation: {x^2\over a^2}+{y^2\over b^2}-1=0.

In general, if there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form

L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ + (lower-order terms)} =0 \,, where L is an elliptic operator.

For example, in three dimensions (x,y,z) :

a\frac{\partial^2 u}{\partial x^2} + b\frac{\partial^2 u}{\partial x\partial y} + c\frac{\partial^2 u}{\partial y^2} + d\frac{\partial^2 u}{\partial y\partial z} + e\frac{\partial^2 u}{\partial z^2} \text{ + (lower-order terms)}= 0,

which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives

a\frac{\partial^2 u}{\partial x^2} + c\frac{\partial^2 u}{\partial y^2} + e\frac{\partial^2 u}{\partial z^2} \text{ + (lower-order terms)}= 0.

This can be compared to the equation for an ellipsoid; {x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.

See also

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