- Lions–Lax–Milgram theorem
In
mathematics , the Lions–Lax–Milgram theorem (or simply Lions’ theorem) is a result infunctional analysis with applications in the study ofpartial differential equation s. It is a generalization of the famousLax-Milgram theorem , which gives conditions under which abilinear function can be "inverted" to show the existence and uniqueness of aweak solution to a givenboundary value problem . The result is named after the mathematiciansJacques-Louis Lions ,Peter Lax andArthur Milgram .tatement of the theorem
Let "H" be a
Hilbert space and "V" anormed space . Let "B" : "H" × "V" → R be a continuous, bilinear function. Then the following are equivalent:* (coercivity) for some constant "c" > 0,
::
* (existence of a "weak inverse") for each
continuous linear functional "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 "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’ 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' theorem, consider the
heat equation in "n" spatial dimensions ("x") and one time dimension ("t")::
where Δ denotes the
Laplace operator . Two questions arise immediately: on what domain inspacetime 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 approximation s) on each "time slice" {"t"} × 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 thepolar ice cap , one must take account of the changing shape of the volume of ice as itevaporate s and/oriceberg s 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’ theorem.ee also
*
Babuška-Lax-Milgram theorem References
* cite book
last = Showalter
first = Ralph E.
title = Monotone operators in Banach space and nonlinear partial differential equations
series = Mathematical Surveys and Monographs 49
publisher = American Mathematical Society
location = Providence, RI
year = 1997
pages = pp. xiv+278
isbn = 0-8218-0500-2 MathSciNet|id=1422252 (chapter III)
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