Weyl's lemma (Laplace equation), the Glossary

Table of Contents

1. 21 relations: Analytic function, Convolution, Distribution (mathematics), Elliptic partial differential equation, Harmonic function, Hermann Weyl, Hypoelliptic operator, Laplace operator, Laplace's equation, Lars Gårding, Lars Hörmander, Locally integrable function, Mathematics, Mollifier, Null set, Open set, Smoothness, Spaces of test functions and distributions, Support (mathematics), Wave equation, Weak solution.

2. Harmonic functions
3. Lemmas in analysis

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series.

See Weyl's lemma (Laplace equation) and Analytic function

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g).

See Weyl's lemma (Laplace equation) and Convolution

Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis.

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Elliptic partial differential equation

Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Weyl's lemma (Laplace equation) and elliptic partial differential equation are partial differential equations.

See Weyl's lemma (Laplace equation) and Elliptic partial differential equation

Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac. Weyl's lemma (Laplace equation) and harmonic function are harmonic functions.

See Weyl's lemma (Laplace equation) and Harmonic function

Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher.

See Weyl's lemma (Laplace equation) and Hermann Weyl

Hypoelliptic operator

In the theory of partial differential equations, a partial differential operator P defined on an open subset is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty (smooth), u must also be C^\infty. Weyl's lemma (Laplace equation) and hypoelliptic operator are partial differential equations.

See Weyl's lemma (Laplace equation) and Hypoelliptic operator

Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. Weyl's lemma (Laplace equation) and Laplace operator are harmonic functions.

See Weyl's lemma (Laplace equation) and Laplace operator

Laplace's equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. Weyl's lemma (Laplace equation) and Laplace's equation are harmonic functions.

See Weyl's lemma (Laplace equation) and Laplace's equation

Lars Gårding

Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in Sweden 1952–1984. Together with Marcel Riesz, he was a thesis advisor for Lars Hörmander.

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Lars Hörmander

Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations".

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Locally integrable function

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition.

See Weyl's lemma (Laplace equation) and Locally integrable function

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Mollifier

In mathematics, mollifiers (also known as approximations to the identity) are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.

See Weyl's lemma (Laplace equation) and Mollifier

Null set

In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero.

See Weyl's lemma (Laplace equation) and Null set

Open set

In mathematics, an open set is a generalization of an open interval in the real line.

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Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.

See Weyl's lemma (Laplace equation) and Smoothness

Spaces of test functions and distributions

In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions.

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Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero.

See Weyl's lemma (Laplace equation) and Support (mathematics)

Wave equation

The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).

See Weyl's lemma (Laplace equation) and Wave equation

Weak solution

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense.

See Weyl's lemma (Laplace equation) and Weak solution

See also

Harmonic functions

• Bôcher's theorem
• Cauchy–Riemann equations
• Differential forms on a Riemann surface
• Dirichlet's principle
• Edmund Schuster
• Harmonic conjugate
• Harmonic coordinates
• Harmonic function
• Harmonic map
• Harmonic morphism
• Harnack's inequality
• Harnack's principle
• Hilbert transform
• Kellogg's theorem
• Kelvin transform
• Laplace operator
• Laplace's equation
• Maximum principle
• Newtonian potential
• Pluriharmonic function
• Poisson kernel
• Positive harmonic function
• Schwarz alternating method
• Schwarz reflection principle
• Weakly harmonic function
• Weyl's lemma (Laplace equation)

Lemmas in analysis

• Aubin–Lions lemma
• Auerbach's lemma
• Borel's lemma
• Bramble–Hilbert lemma
• Céa's lemma
• Calderón–Zygmund lemma
• Closed and exact differential forms
• Cotlar–Stein lemma
• Ehrling's lemma
• Estimation lemma
• Fatou's lemma
• Fundamental lemma of the calculus of variations
• Grönwall's inequality
• Halanay inequality
• Itô's lemma
• Jordan's lemma
• Lebesgue's lemma
• Lions–Magenes lemma
• Malliavin's absolute continuity lemma
• Mazur's lemma
• Morse–Palais lemma
• Oka's lemma
• Poincaré lemma
• Pugh's closing lemma
• Riemann–Lebesgue lemma
• Riesz's lemma
• Rising sun lemma
• Sard's theorem
• Schwarz lemma
• Spijker's lemma
• Stechkin's lemma
• Stewart–Walker lemma
• Watson's lemma
• Weyl's lemma (Laplace equation)
• Wiener's lemma

References

[1] https://en.wikipedia.org/wiki/Weyl's_lemma_(Laplace_equation)

Also known as Weyl lemma (Laplace equation).