“Friedrichs's inequality”的意思、由来-开放百科全书

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Friedrichs's inequality

释义

Statement of the inequality
See also
References

In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the L^p norm of a function using L^p bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent. Friedrichs's inequality is a general case of the Poincaré–Wirtinger inequality which deals with the case k = 1.

Statement of the inequality

Let be a bounded subset of Euclidean space with diameter . Suppose that lies in the Sobolev space , i.e., and the trace of on the boundary is zero. Then

In the above

denotes the L^p norm;
α = (α₁, ..., α_n) is a multi-index with norm |α| = α₁ + ... + α_n;
D^αu is the mixed partial derivative

References

{{cite book |first=Karel |last=Rektorys |chapter=The Friedrichs Inequality. The Poincaré inequality |title=Variational Methods in Mathematics, Science and Engineering |location=Dordrecht |publisher=Reidel |origyear=1977 |edition=2nd |year=2001 |isbn=1-4020-0297-1 |pages=188–198 |chapterurl=https://books.google.com/books?id=Kob1CAAAQBAJ&pg=PA188 }}

2 : Sobolev spaces|Inequalities

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Statement of the inequality

See also

References