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Generalizing the maximum principle for harmonic functions which was already known to Gauss in 1839, Eberhard Hopf proved in 1927 that if a function satisfies a second order partial differential inequality of a certain kind in a domain of R n and attains a maximum in the domain then the function is constant. The simple idea behind Hopf's proof ...
The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the ...
The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained.
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function (i.e, | | < ()) on an unbounded domain when an additional (usually mild) condition constraining the ...
In one of his earliest works, Nirenberg adapted Hopf's proof to second-order parabolic partial differential equations, thereby establishing the strong maximum principle in that context. As in the earlier work, such a result had various uniqueness and comparison theorems as corollaries. Nirenberg's work is now regarded as one of the foundations ...
For bounded domains, the Dirichlet problem can be solved using the Perron method, which relies on the maximum principle for subharmonic functions. This approach is described in many text books. [ 2 ] It is not well-suited to describing smoothness of solutions when the boundary is smooth.
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set , every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect to set inclusion.
Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted ...