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Harmonic functions satisfy the following maximum principle: if K is a nonempty compact subset of U, then f restricted to K attains its maximum and minimum on the boundary of K. If U is connected , this means that f cannot have local maxima or minima, other than the exceptional case where f is constant .
This is the weak maximum principle for harmonic functions. This does not, by itself, rule out the possibility that the maximum of u is also attained somewhere on M. That is the content of the "strong maximum principle," which requires further analysis. The use of the specific function above was very inessential. All that mattered was to have a ...
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 ...
In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward ...
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.
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following: The fundamental theorem of algebra . Schwarz's lemma , a result which in turn has many generalisations and applications in complex analysis.
Formally, the definition can be stated as follows. Let be a subset of the Euclidean space and let : {} be an upper semi-continuous function.Then, is called subharmonic if for any closed ball (,) ¯ of center and radius contained in and every real-valued continuous function on (,) ¯ that is harmonic in (,) and satisfies () for all on the boundary (,) of (,), we have () for all (,).
The operator takes a locally integrable function f : R d → C and returns another function Mf. For any point x ∈ R d, the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x. Formally,