Poincaré inequality

A Poincare inequality controls a function by its derivatives after the constant ambiguity has been removed. For example, on a bounded connected Lipschitz domain,

Suppose $u\in W^{1,2}(B_R)$ vanishes on a measurable subset of measure at least $\theta |B_R|$, with $\theta >0$. Writing $m$ for its mean, the zero set gives $\theta |B_R|m^2\le {\int }_{B_R}|u-m{|}^2$. Thus $\int u^2\le (1+{\theta }^{-1})\int |u-m{|}^2$, and the Poincare-Wirtinger inequality supplies the displayed estimate. An elementary proof of that inequality on a ball bounds differences along line segments and splits the segment parameter at $1/2$, giving the valid explicit constant $C(n,\theta )=2^{n+1}(1+{\theta }^{-1})$. The change of scale from the unit ball multiplies the constant by $R^2$.

For $n\ge 2$, a round annulus $A_R=B_{2R}\setminus B_R$ is connected and obeys ${\int }_{A_R}|v-v_{A_R}{|}^2\le C(n)R^2{\int }_{A_R}|\nabla v{|}^2$. Scaling reduces the claim to a fixed annulus. In dimension one the annulus has two components, so the inequality fails with a single average; one needs a separate average on each component.

On a bounded connected Lipschitz domain $U$, every $u\in H^1(U)$ satisfiesEquivalently, the gradient norm is equivalent to the $H^1$ norm on the mean-zero Sobolev space.

For the cube $Q=(0,L{)}^n$ and $u\in H^1(Q)$,This follows from the pairwise-difference identityand a coordinate-by-coordinate path from $x$ to $y$. The Cauchy-Schwarz inequality bounds the squared sum of its $n$ increments by $n$ times the sum of their squares, while the one-dimensional fundamental theorem of calculus bounds each integrated increment by $L^{n+2}\Vert D_{i}u{\Vert }_2^2$. Division by $2|Q|=2L^n$ gives the stated constant.

The mean-zero Sobolev space is the closed subspaceOn a bounded connected Lipschitz domain, the Poincare-Wirtinger inequality makes $\Vert \nabla u{\Vert }_2$ an equivalent Hilbert norm on this space.

If $U$ is bounded and connected and a boundary portion ${\Gamma }_D$ has positive surface measure, thenfor every $v\in H^1(U)$ whose trace vanishes on ${\Gamma }_D$. Otherwise a normalized counterexample sequence, the Rellich-Kondrashov compactness theorem, and continuity of the trace would produce a nonzero constant with zero trace on ${\Gamma }_D$.

On a bounded connected Lipschitz domain $U$,for every $u\in H^1(U)$. If this failed, the Rellich-Kondrachov compactness theorem would produce a normalized strong $L^2$ limit with zero gradient, hence a constant function; continuity of the Sobolev trace theorem would force that constant to vanish, contradicting its unit norm.

For a probability density function $w$ on a domain, a weighted Poincare inequality has the formThe optimal $\lambda$ is the spectral gap of the associated weighted diffusion operator.

For $w=e^{-\Phi }$ and $g=h{e}^{-\Phi /2}$, integration by parts givesThis conjugates the weighted Dirichlet energy to a Schrödinger operator with effective potential $|{\Phi }^{\prime }{|}^2/4-{\Phi }^{\prime \prime }/2$.