Friedrichs's inequality

In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp 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 Lp norm;
  • α = (α1, ..., αn) is a multi-index with norm |α| = α1 + ... + αn;
  • Dαu is the mixed partial derivative
gollark: ???
gollark: I don't think you need to compile the kernel for serial IO.
gollark: Broadly speaking, you have a parser which turns the text into abstract syntax trees representing the code (`1 + 1` goes to `Operator("+", 1, 1)` or something, for example), then you generate structures for all the various functions and whatever and check things for validity, then turn those into output code.
gollark: Compilers are generally quite complex. I forgot what the best resources for them were.
gollark: Or database or something, yes.

See also

References

  • Rektorys, Karel (2001) [1977]. "The Friedrichs Inequality. The Poincaré inequality". Variational Methods in Mathematics, Science and Engineering (2nd ed.). Dordrecht: Reidel. pp. 188–198. ISBN 1-4020-0297-1.
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