Lions–Lax–Milgram theorem
In
Statement of the theorem
Let H be a
- (coercivity) for some constant c > 0, [citation needed]
- (existence of a "weak inverse") for each continuous linear functionalf ∈ 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
- 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 n spatial dimensions (x) and one time dimension (t):
where Δ denotes the Laplace operator. Two questions arise immediately: on what domain in spacetime 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
See also
References
- Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. pp. xiv+278. MR1422252(chapter III)