Elliptic partial differential equation

Summary

An elliptic partial differential equation is one whose principal symbol vanishes for no nonzero covector. For second-order equations this means that the matrix of second-order coefficients is definite. Elliptic equations model equilibrium states, such as steady temperatures, electrostatic potentials and membranes at rest. Their solutions are as smooth as the data allow, and boundary value problems are the natural problems for them. See the definition.