In
mathematics
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 ...
, the Lions–Lax–Milgram theorem (or simply Lions's theorem) is a result in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
with applications in the study of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s. It is a generalization of the famous
Lax–Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...
, which gives conditions under which a
bilinear function can be "inverted" to show the existence and uniqueness of a
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 ...
to a given
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
. The result is named after the mathematicians
Jacques-Louis Lions
Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture ...
,
Peter Lax
Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.
Lax has made important contributions to integrable systems, fluid dyn ...
and
Arthur Milgram
Arthur Norton Milgram (3 June 1912, in Philadelphia – 30 January 1961) was an American mathematician. He made contributions in functional analysis, combinatorics, differential geometry, topology, partial differential equations, and Galois theor ...
.
Statement of the theorem
Let ''H'' be a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
and ''V'' a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
. Let ''B'' : ''H'' × ''V'' → R be a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, bilinear function. Then the following are equivalent:
* (
coercivity
Coercivity, also called the magnetic coercivity, coercive field or coercive force, is a measure of the ability of a ferromagnetic material to withstand an external magnetic field without becoming demagnetized. Coercivity is usually measured in ...
) for some constant ''c'' > 0,
::
* (existence of a "weak inverse") for each
continuous linear functional In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear ...
''f'' ∈ ''V''
∗, there is an element ''h'' ∈ ''H'' such that
::
Related results
The Lions–Lax–Milgram theorem can be applied by using the following result, the hypotheses of which are quite common and easy to verify in practical applications:
Suppose that ''V'' is
continuously embedded In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defi ...
in ''H'' and that ''B'' is ''V''-elliptic, i.e.
* for some ''c'' > 0 and all ''v'' ∈ ''V'',
::
* for some ''α'' > 0 and all ''v'' ∈ ''V'',
::
Then the above coercivity condition (and hence the existence result) holds.
Importance and applications
Lions's generalization is an important one since it allows one to tackle boundary value problems beyond the Hilbert space setting of the original Lax–Milgram theory. To illustrate the power of Lions's theorem, consider the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
in ''n'' spatial dimensions (''x'') and one time dimension (''t''):
:
where Δ denotes the
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 ...
. Two questions arise immediately: on what domain in
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
is the heat equation to be solved, and what boundary conditions are to be imposed? The first question — the shape of the domain — is the one in which the power of the Lions–Lax–Milgram theorem can be seen. In simple settings, it suffices to consider ''cylindrical domains'': i.e., one fixes a spatial region of interest, Ω, and a maximal time, ''T'' ∈(0, +∞], and proceeds to solve the heat equation on the "cylinder"
:
One can then proceed to solve the heat equation using classical Lax–Milgram theory (and/or Galerkin approximations) on each "time slice" × Ω. This is all very well if one only wishes to solve the heat equation on a domain that does not change its shape as a function of time. However, there are many applications for which this is not true: for example, if one wishes to solve the heat equation on the polar ice cap, one must take account of the changing shape of the volume of ice as it evaporates and/or icebergs break away. In other words, one must at least be able to handle domains ''G'' in spacetime that do not look the same along each "time slice". (There is also the added complication of domains whose shape changes according to the solution ''u'' of the problem itself.) Such domains and boundary conditions are beyond the reach of classical Lax–Milgram theory, but can be attacked using Lions's theorem.
See also
*
Babuška–Lax–Milgram theorem In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given ...
References
* (chapter III)
{{DEFAULTSORT:Lions-Lax-Milgram theorem
Partial differential equations
Theorems in functional analysis