Partial differential equations Definition

Elliptic partial differential equation

Builds on Linear PDE Principal symbol Positive definite matrix

Definition

Let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$ be a linear differential operator of order $m$ on an open set $\Omega \subseteq \R^n$ , with principal symbol $\sigma_m(P)$ . The operator $P$ is elliptic at $x \in \Omega$ if $\sigma_m(P)(x, \xi) \ne 0$ for every $\xi \in \R^n \setminus \set{0}$ . It is elliptic in $\Omega$ if it is elliptic at every point of $\Omega$ . The equation $P u = f$ is then an elliptic partial differential equation.

For a second-order operator $\sum_{i,j} a_{ij}(x) \partial_i \partial_j + \sum_i b_i(x) \partial_i + c(x)$ with real symmetric $(a_{ij})$ , this means that the matrix $(a_{ij}(x))$ is positive definite or negative definite at each $x$ . It is uniformly elliptic in $\Omega$ if there is a constant $\theta > 0$ with $\sum_{i,j=1}^{n} a_{ij}(x) \, \xi_i \xi_j \ge \theta \abs{\xi}^2 \quad \text{for all } x \in \Omega, \ \xi \in \R^n.$

Remarks

Example: the Laplacian $\Delta = \sum_i \partial_i^2$ has principal symbol $\abs{\xi}^2$ , which is nonzero for $\xi \ne 0$ , so it is elliptic; it is uniformly elliptic with $\theta = 1$ . Example: the Cauchy–Riemann operator $\partial_x + i \partial_y$ on $\R^2$ has symbol $\xi_1 + i \xi_2$ , which vanishes only at $\xi = 0$ , so it is elliptic of order $1$ . Non-example: the wave operator $\partial_t^2 - \partial_x^2$ has symbol $\tau^2 - \xi^2$ , which vanishes at $(\tau, \xi) = (1, 1)$ , so it is not elliptic. If all coefficients are real and $n \ge 2$ , an elliptic operator has even order $m$ . Indeed $\sigma_m(x, -\xi) = (-1)^m \sigma_m(x, \xi)$ , and a real continuous function that does not vanish on the connected unit sphere cannot change sign. For nonlinear second-order equations $F(D^2 u, D u, u, x) = 0$ , ellipticity is a monotonicity condition of $F$ in the matrix variable $D^2 u$ .

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Linear PDE concept
Positive definite matrix concept
Principal symbol concept
Elliptic partial differential equation definition

Tag	`3BYHP9E9VF`
URI	`m2://3BYHP9E9VF`
Kind	definition
Writing	W0 stub
MSC	35J15, 35J30, 47F10
Defines	Elliptic partial differential equation
Uses	Linear PDE, Principal symbol, Positive definite matrix

Elliptic partial differential equation

Definition

Remarks

Reading path

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