Category:Elliptic

This category contains all control problems which are governed by an elliptic partial differential equation.

A second order linear partial differential equation can be written as ${\displaystyle {\displaystyle \sum _{i,j=1}^n}{a}_{ij}\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+\text{lower-order terms}=0}$.

If the matrix ${\displaystyle A=(a_{ij}{)}_{ij}}$ is positive or negative definite, the partial differential equation is called elliptic.

An example is the Poisson's equation: ${\displaystyle -\mathrm{\Delta }u=f}$, where ${\displaystyle \mathrm{\Delta }}$ denotes the Laplace operator, ${\displaystyle u}$ is the unknown, and the function ${\displaystyle f}$ is given.