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Lax-Milgram Theorem


In functional analysis, the Lax-Milgram theorem is a sort of representation theorem for bounded linear functionals on a Hilbert space H. The result is of tantamount significance in the study of function spaces and partial differential equations.

Let phi be a bounded coercive bilinear form on a Hilbert space H. The Lax-Milgram theorem states that, for every bounded linear functional f on H, there exists a unique x_f in H such that

 f(x)=phi(x,x_f)

for all x in H.

It is worth noting that the Lax-Milgram theorem follows immediately as a corollary of the Stampacchia theorem. One version of the Stampacchia theorem says that, under the assumptions above, any function f in H necessarily corresponds to a unique function u in H for which the inequality

 phi(u,v-u)>=<f,v-u>_H

is satisfied for all functions v in H where here, <·,·>_H denotes the inner product on H.


See also

Stampacchia Theorem

Portions of this entry contributed by Christopher Stover

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References

Debnath, L. and Mikusiński, P. Introduction to Hilbert Spaces with Applications. San Diego, CA: Academic Press, 1990.Dimitrios, K. "The Stampacchia and Lax-Milgram Theorems and Applications." http://www.stat-athens.aueb.gr/gr/master/sumschool/files/Kravvaritis.pdf.Monteillet, A. "A Theorem of Stampacchia." http://aurelien.monteillet.com/Stages/Stampacchia-anglais.pdf.Stampacchia, G. "Équations elliptiques du second ordre à coefficients discontinus." Séminaire Jean Leray 3, 1-77, 1963-1964. http://www.numdam.org/item?id=SJL_1963-1964___3_1_0.Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.

Referenced on Wolfram|Alpha

Lax-Milgram Theorem

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Lax-Milgram Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Lax-MilgramTheorem.html

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