Kellogg's theorem is a pair of related results in the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

study of the regularity of

harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \fr ...

on sufficiently smooth domains by

Oliver Dimon Kellogg Oliver Dimon Kellogg (10 July 1878 – 27 August 1932) was an American mathematician. His father, Day Otis Kellogg, was a professor of literature at the University of Kansas and editor of the American edition of the ''Encyclopædia Britannica''. ...

. In the first version, it states that, for

k \geq 2

, if the domain's boundary is of class

C^k

and the ''k''-th derivatives of the boundary are Dini continuous, then the harmonic functions are uniformly

C^k

as well. The second, more common version of the theorem states that for domains which are

C^

, if the boundary data is of class

C^

, then so is the harmonic function itself. Kellogg's method of proof analyzes the representation of harmonic functions provided by the

Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriva ...

, applied to an interior tangent sphere. In modern presentations, Kellogg's theorem is usually covered as a specific case of the boundary Schauder estimates for

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...

Sources

* * Harmonic functions Potential theory {{Mathanalysis-stub

See also

Sources