Parametrix

In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator. A parametrix for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good. It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it. It is useful to review what a fundamental solution for a differential operator P(D) with constant coefficients is: it is a distribution u on Rn such that in the weak sense, where δ is the Dirac delta distribution. In a similar way, a parametrix for a variable coefficient differential operator P(x,D) is a distribution u such that where ω is some C ∞ function with compact support. The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic pseudodifferential operators with variable coefficient, since for such operators over appropriate domains a parametrix can be shown to exist, can be somewhat easily constructed and be a smooth function away from the origin. Having found the analytic expression of the parametrix, it is possible to compute the solution of the associated fairly general elliptic partial differential equation by solving an associated Fredholm integral equation: also, the structure itself of the parametrix reveals properties of the solution of the problem without even calculating it, like its smoothness and other qualitative properties. More generally, if L is any pseudodifferential operator of order p, then another pseudodifferential operator L+ of order –p is called a parametrix for L if the operators are both pseudodifferential operators of negative order. The operators L and L+ will admit continuous extensions to maps between the Sobolev spaces Hs and Hs+k. On a compact manifold, the differences above are compact operators. In this case the original operator L defines a Fredholm operator between the Sobolev spaces.