In
mathematics, Weyl's lemma, named after
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
, states that every
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 precisel ...
of
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. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
is a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
solution. This contrasts with the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seis ...
, for example, which has weak solutions that are not smooth solutions. Weyl's lemma is a special case of
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 ...
or
hypoelliptic regularity.
Statement of the lemma
Let
be an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
of
-dimensional Euclidean space
, and let
denote the usual
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. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
. Weyl's lemma states that if a
locally integrable 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. The importance of such functions lie ...
function
is a weak solution of Laplace's equation, in the sense that
:
for every
smooth test function
with
compact support
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. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
, then (up to redefinition on a set of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
)
is smooth and satisfies
pointwise in
.
This result implies the interior regularity of harmonic functions in
, but it does not say anything about their regularity on the boundary
.
Idea of the proof
To prove Weyl's lemma, one
convolves the function
with an appropriate
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) fu ...
and shows that the mollification
satisfies Laplace's equation, which implies that
has the mean value property. Taking the limit as
and using the properties of mollifiers, one finds that
also has the mean value property, which implies that it is a smooth solution of Laplace's equation. Alternative proofs use the smoothness of the fundamental solution of the Laplacian or suitable a priori elliptic estimates.
Generalization to distributions
More generally, the same result holds for every
distributional solution of Laplace's equation: If
satisfies
for every
, then
is a regular distribution associated with a smooth solution
of Laplace's equation.
Connection with hypoellipticity
Weyl's lemma follows from more general results concerning the regularity properties of elliptic or hypoelliptic operators.
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". Hörmander was awarded the Fields Medal ...
, ''The Analysis of Linear Partial Differential Operators I'', 2nd ed., Springer-Verlag (1990), p.110 A linear partial differential operator
with smooth coefficients is hypoelliptic if the
singular support of
is equal to the singular support of
for every distribution
. The Laplace operator is hypoelliptic, so if
, then the singular support of
is empty since the singular support of
is empty, meaning that
. In fact, since the Laplacian is elliptic, a stronger result is true, and solutions of
are
real-analytic.
Notes
References
*
*{{cite book , first=Elias , last=Stein , authorlink=Elias Stein, year=2005 , title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces , publisher=Princeton University Press , isbn=0-691-11386-6
Lemmas in analysis
Partial differential equations
Harmonic functions