In the theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
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
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors clai ...
.
''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 In mathematics, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms a ...
given by , and the estimates given by De Giorgi and Nash for second order elliptic or
parabolic equations in many variables, in their respective solutions to
Hilbert's nineteenth problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less ...
.
References
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Partial differential equations
A priori
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