Kellogg's Theorem

	Kellogg's Theorem Kellogg's theorem is a pair of related results in the mathematical study of the regularity of harmonic functions on sufficiently smooth domains by Oliver Dimon Kellogg. 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, 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 partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 1895 Oliver Kellogg began his undergraduate study at Princeton University, where he earned his master's degree in 1900. With a John S. Kennedy stipend he first studied at the Humboldt University of Berlin and then in 1901/1902 at Georg-August-Universität Göttingen. At Göttingen in 1902 he earned his PhD with a thesis ''Zur Theorie der Integralgleichungen und des Dirichlet'schen Prinzips'' under the direction of David Hilbert. After completing his thesis, Kellogg became an instructor at Princeton and from 1905 at the University of Missouri, where he became a professor in 1910. In World War I he was a scientific advisor at the Coast Guard Academy in New London, Connecticut, where he worked on submarine detection. Kellogg became a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Dini Continuity In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous. Definition Let X be a compact subset of a metric space (such as \mathbb^n), and let f:X\rightarrow X be a function from X into itself. The modulus of continuity of f is :\omega_f(t) = \sup_ d(f(x),f(y)). The function f is called Dini-continuous if :\int_0^1 \frac\,dt < \infty. An equivalent condition is that, for any $\theta \in (0,1)$ , : $\sum_^\infty \omega_f(\theta^i a) < \infty$ where $a$ is the diameter of $X$ . See also * Dini test ~~[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]~~
	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 derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to ''n''-dimensional problems. Two-dimensional Poisson kernels On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\right), \ \ \ 0 \le r < 1. This can be thought of in two ways: either as a function of ''r'' and ''θ'', or as a family of functions of ''θ'' indexed by ''r''. If $D = \$ is the open [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Schauder Estimates In mathematics, 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 that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates. There is both an ''interior'' result, giving a Hölder condition for the solution in interior domains away from the boundary, and a ''boundary'' result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well. The Schauder estimates are a necessary precondition to using the method of continuity to pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Elliptic Operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transactions Of The American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * '' Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

See also