Harnack's principle

Harnack's principle states that an increasing sequence of positive harmonic functions on a domain either converges locally uniformly to a harmonic function or diverges everywhere.

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Harnack inequality: An inequality that bounds the values of a positive harmonic function at different points within a domain. It supplies the local bounds that make an increasing sequence converge uniformly on compact subsets.
Positive harmonic function: A harmonic function that takes strictly positive values throughout its domain. Positivity is essential to the estimates that prevent mixed finite and infinite behavior.
Harnack convergence theorem: A theorem that an increasing sequence of positive harmonic functions either converges locally uniformly to a harmonic function or diverges everywhere. This named convergence result is the direct operational form of Harnack's principle.
Carl Gustav Jacob Jacobi: A German mathematician who made foundational contributions to analysis, mechanics, and number theory. His work helped establish the nineteenth-century study of harmonic and elliptic functions.
Harmonic function: A twice-differentiable function whose Laplacian vanishes throughout its domain. These are the functions in the sequence and the possible finite limit.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. The theorem is stated on a domain, an open connected set in Euclidean space.
Normal family: A family of functions for which every sequence has a subsequence converging uniformly on compact subsets, under suitable conditions. Harnack bounds help establish subsequential compactness for families of positive harmonic functions.
Friedrich Wilhelm Bessel: A German mathematician and astronomer known for foundational work on Bessel functions and mathematical astronomy. His era's analysis helped shape the setting in which harmonic-function theory developed.
Local uniform convergence: Convergence that is uniform on every compact subset of a space. This is the mode of convergence guaranteed when the sequence does not diverge everywhere.
Laplacian: The differential operator that sums a function's second partial derivatives. A harmonic function is defined by the vanishing of this operator.