Poincaré Inequality

An inequality that bounds the L^p norm of a function (after removing a rigid mode such as mean or boundary trace) by the L^p norm of its gradient on a domain, exhibiting that gradients control function oscillation up to constants.

An inequality that bounds the L2 norm of a function (after subtracting an appropriate mean or enforcing boundary conditions) by a constant times the L2 norm of its gradient on a domain, quantifying how oscillation is controlled by first derivatives.