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The singular equation $2x{w}^{\prime \prime }-w^{\prime }+\lambda x^{2}w=0$ becomes $W^{\prime \prime }+(2\lambda /9)W=0$ under $z=x^{3/2}$, $w(x)=W(z)$. Its zero-endpoint finite-energy quotient transforms as $2{\int }_0^1{x}^{-1/2}(w^{\prime }{)}^{2}dx/{\int }_0^1{x}^{1/2}{w}^{2}dx=(9/2){\int }_0^1(W^{\prime }{)}^{2}dz/{\int }_0^1{W}^{2}dz$. The Poincaré inequality therefore proves the sharp minimum $9{\pi }^2/2$, attained by $w=\sin (\pi x^{3/2})$. This coordinate change turns an apparently singular weighted variational problem into the usual interval energy problem. The admissible eigenfunction is $O(x^{3/2})$, making the integration-by-parts boundary term $w{x}^{-1/2}{w}^{\prime }$ vanish and both energy integrals converge.

For a nonzero trial function with $y(0)=y(L)=0$ and $y(L-x)=y(x)$, substituting $L-x$ for $x$ proves ${\int }_0^{L}x{y}^{2}dx=(L/2){\int }_0^L{y}^{2}dx$. This gives the displayed Rayleigh quotient. The Poincaré inequality on an interval bounds it below by $2{\pi }^2/L^3$, with equality for a nonzero multiple of $\sin (\pi x/L)$. Thus this sine is the best reflection-symmetric trial for $-y^{\prime \prime }=\lambda xy$, while the unrestricted lowest eigenfunction need not be reflection-symmetric. The polynomial $x(L-x)$ gives $20/L^3$, only a factor $10/{\pi }^2$ above the sine's value. Both are upper bounds on the unrestricted lowest eigenvalue by the Rayleigh-Ritz method.

For a globally Lipschitz nonlinearity of constant $ϵ$, the map taking $v$ to the zero-Dirichlet Poisson solution with load $f(v)$ is a contraction in the gradient norm if the displayed inequality holds. Poincaré gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear Poisson equation gives another factor $C_P$ back to the gradient norm. The condition yields a unique weak solution without requiring $f(0)=0$. With the optimal Poincaré constant it reads $ϵ<{\lambda }_1$.

On $(0,L)$ with zero Dirichlet boundary conditions, the normalized trial $\sqrt{2/L}\sin (\pi x/L)$ has kinetic Rayleigh quotient ${\pi }^2/L^2$ and mean position $L/2$ by reflection symmetry. The Rayleigh-Ritz variational principle therefore gives the displayed upper bound for a real linear potential. For $ϵ\ge 0$ the lower bound ${\lambda }_1\ge {\pi }^2/L^2$ follows from the Poincaré inequality. For $ϵ\ne 0$ the trial is not an exact eigenfunction, so the upper bound is strict.

A skew-coupled elliptic system has zeroth-order couplings with opposite signs. For the displayed example with homogeneous Dirichlet boundary conditions, its weak formulation on $V=H_0^1(\Omega )\times H_0^1(\Omega )$ has the bilinear formThe coupling cancels for equal arguments, so $b((p,q),(p,q))=\Vert \nabla p{\Vert }_0^2+\Vert \nabla q{\Vert }_0^2$. On a domain satisfying the Poincaré inequality, this is a coercive bilinear form. With the product first-order Sobolev norm, the Cauchy-Schwarz inequality applied to $(\nabla p,\nabla q,-q,p)$ and $(\nabla v,\nabla z,v,z)$ gives $|b(P,T)|\le \Vert P{\Vert }_V\Vert T{\Vert }_V$. The Lax-Milgram theorem therefore applies even though the bilinear form is not symmetric. Eliminating $w=-\Delta u$ recovers ${\Delta }^{2}u+u=f$, involving the biharmonic operator.

For $1\le p<\infty$, an inhomogeneous Sobolev space norm is $\Vert u{\Vert }_{W^{k,p}(D)}=({\sum }_{|\alpha |\le k}\Vert D^{\alpha }u{\Vert }_{L^p(D)}^p{)}^{1/p}$, using weak derivatives. In particular, $\Vert u{\Vert }_{H^1(D)}^2={\int }_D(|u{|}^2+|\nabla u{|}^2)$. The gradient seminorm alone defines a different completion, the Dirichlet energy space, unless a suitable Poincaré inequality makes the two norms equivalent on the chosen zero-boundary space.

For zero-mean fields in a shearing box, Fourier modes have wavevectors $(2\pi n/L_x+2At2\pi m/L_y,2\pi m/L_y,2\pi p/L_z)$. A nonzero azimuthal or vertical index supplies at least $2\pi /L$, where $L$ is the largest box side; otherwise the nonzero radial index does. Parseval identity gives the displayed Poincaré inequality uniformly in time.

The log-cosh derivative operator is strictly monotone on $H^1$ but does not have $⟨A(v),v⟩/\Vert v{\Vert }_{H^1}\to \infty$. Constant functions $v=t$ make the ratio approach $|\Omega {|}^{1/2}$. Restricting to $H_0^1$ removes the uncontrolled constants: Poincaré gives $⟨A(v),v⟩\ge \Vert v{\Vert }_{H^1}^2/(1+C_P^2)$, which proves coercivity. Strict monotonicity and large-norm coercivity are distinct properties.

For a bounded domain, bounded symmetric uniformly elliptic $A$, $q\le 0$, and $f\in L^2$, minimizing $J$ on $\psi +H_0^1$ solves $\mathrm{div}(A\nabla u)+qu=f$. The nonnegative quadratic terms are weakly lower semicontinuous, and Poincaré inequality makes the gradient norm coercive on the zero-boundary part. The first variation gives the weak equation with the forcing sign displayed above. If $q$ is positive, coercivity may fail at a Dirichlet eigenvalue; orthogonality to the homogeneous kernel is then needed for solvability.

A second-order operator with coefficient matrix $A(x)=(a^{ij}(x))$ is uniformly elliptic when some $\lambda >0$ satisfies $a^{ij}(x){\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ everywhere.

For $\overline{w{a}^{\prime }}=-{\eta }_t{B}_{0}f$, $⟨f⟩=1$ and positive diffusivities, the Zeldovich planar flux balance gives $G=({\eta }_t/\eta )B_0^2[1-({\eta }_t/\eta )\mathrm{Var}(f)]$, where $G=⟨|\nabla a^{\prime }{|}^2⟩$. The Cauchy-Schwarz inequality and a Poincaré inequality $⟨a^{\prime 2}⟩\le d^{2}G/{\pi }^2$ give ${\eta }_t^2\mathrm{Var}(f)\le ⟨w^2⟩d^{2}s(1-s)/{\pi }^2$, with $s=({\eta }_t/\eta )\mathrm{Var}(f)\in [0,1]$. Maximizing $s(1-s)$ gives the displayed bound. Large turbulent-to-microscopic diffusivity therefore requires $f$ to be close to one in mean square, without guaranteeing pointwise uniformity.

On $H=\{v\in H^1(0,1):v(0)=0\}$, the formis bounded and coercive. The Hardy inequality on an interval controls its singular term, while the one-sided Poincaré inequality controls the first-order term. Thus the Lax-Milgram theorem gives a unique weak solution for every functional $v\mapsto {\int }_0^1(f/x)(v/x)$ with $f/x\in L^2(0,1)$.

The zero-trace first-order Sobolev space is the closure of $C_c^{\infty }(\Omega )$ in the $H^1$ norm. On a bounded Lipschitz domain it is the kernel of the Sobolev trace operator. On an interval, its elements have absolutely continuous representatives with both endpoint values zero. The Poincaré inequality makes the derivative norm equivalent to the full Sobolev norm. This is the natural form domain for an elliptic problem with homogeneous Dirichlet boundary conditions.