📂Partial Differential Equations

Elliptic Partial Differential Equations

Definition^{1 2}

Consider the following second-order linear partial differential equation for $u(x,y)$.

$$Au_{xx} + Bu_{xy} + Cu_{yy} + Du_{x} + Eu_{y} + Fu + G = 0\qquad (ABC \ne 0) \tag{1}$$

Here, the coefficients $A, \dots, G$ are functions of $(x,y)$. $\Delta = B^{2} - 4AC$ is called the discriminant. A partial differential equation $(1)$ with a negative discriminant is called an elliptic PDE.

$$(1) \text{ is called elliptic, if } \Delta (x,y) \lt 0.$$

Explanation

In fact, it is rare to call it an elliptic partial differential equation, and it is commonly called [elliptic PDE] directly. The origin of the name is, of course, an ellipse.

If the quadratic curve $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$ satisfies $B^{2} - 4AC \lt 0$, it is an ellipse.

In a narrow sense, it refers to the Poisson equation.

$$\Delta u = -f \qquad (\Delta = 0^{2} - 4(1)(1) = -4)$$