In mathematics, Friedrichs' 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' inequality is a general case of the Poincaré–Wirtinger inequality which deals with the case .
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