A priori estimate
Encyclopedia
In the theory of partial differential equation
s, an a priori estimate (also called an apriori estimate or a priori bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
A priori estimates were introduced and named by , who used them to prove existence of solutions to second order nonlinear elliptic equations in the plane. Some other early influential examples of a priori estimates include the Schauder estimates
given by , and the estimates given by De Giorgi and Nash for second order elliptic or parabolic equations in many variables in their solution to Hilbert's nineteenth problem
.
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s, an a priori estimate (also called an apriori estimate or a priori bound) is an estimate for the size of a solution or its derivatives of a partial differential equation. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem.
A priori estimates were introduced and named by , who used them to prove existence of solutions to second order nonlinear elliptic equations in the plane. Some other early influential examples of a priori estimates include the Schauder estimates
Schauder estimates
In mathematics, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations...
given by , and the estimates given by De Giorgi and Nash for second order elliptic or parabolic equations in many variables in their solution to Hilbert's nineteenth problem
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic.-History:...
.