Functional inequality

A functional inequality bounds quantities associated with an entire class of functions, such as their norms, integrals or derivatives. Its hypotheses, domain, boundary conditions and constant are part of the statement. Poincaré inequalities and Sobolev inequalities are examples that control a function using derivative information.

For an absolutely continuous real function $u$ on $[0,1]$ with $u(0)=0$ and $u^{\prime }\in L^2$, the Cauchy-Schwarz inequality givesIntegrating in $x$ yields ${\int }_0^1|u{|}^2\le \frac{1}{2}{\int }_0^1|u^{\prime }{|}^2$. This constant is sufficient for the argument; no optimality is asserted.

Without a boundary or normalization condition the derivative cannot control a nonzero constant function, whose derivative vanishes. Such inequalities explain how energy estimates can control solutions of partial differential equations and give coercive bounds in variational problems.