In the theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
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.
In mathematical ...
.
''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, and more precisely, in Functional analysis and PDEs, 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 ...
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 by David Hilbert in 1900. 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
Estimate
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
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