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An operator $F$ on an inner-product space is monotone whenfor every $v,w$. The gradient of every differentiable convex function is monotone.

For a convex objective with an $L$-Lipschitz gradient, Nesterov's accelerated gradient method reaches objective error $\delta$ in $O(\sqrt{L{R}^2/\delta })$ iterations when a minimizer lies within distance $R$ of the initial point.

On the unit square, let $g=0$ and take $u_0={\chi }_{\{x_1>0.4\}}$, $u_1={\chi }_{\{x_1>0.6\}}$. Both have one unit-length jump and zero ordinary gradient. Their mean has two unit-length jumps. For squared-error coefficient one, $[E(u_0)+E(u_1)]/2=0.5+\beta$ while $E((u_0+u_1)/2)=0.45+2\beta$, violating convexity when $\beta >0.05$. The Mumford–Shah functional can therefore be nonconvex even though the fixed-edge reconstruction subproblem is strictly convex.

A stationary point is nonisolated if every neighbourhood contains other stationary points. For a twice differentiable function, a regular smooth curve of stationary points through a point forces its Hessian matrix to annihilate the curve's nonzero tangent: differentiate the vanishing gradient along the curve. Thus the Hessian matrix has a zero eigenvalue there, but that alone need not determine whether the point is a local minimum or local maximum.

For interaction $V=u\int {\varphi }^4/4!$, split $\varphi =\phi +\eta$ into slow and fast Fourier modes, with a centered fast Gaussian measure. The first cumulant expansion term is $⟨V⟩=(u/24)\int [{\phi }^4+6G_{>}(0){\phi }^2+3G_{>}(0{)}^2]$ by Wick theorem. The constant shifts the free energy, the quadratic term gives $\delta r=u{G}_{>}(0)/2$, and there is no gradient correction at this order. Rescaling gives $r^{\prime }=b^2[r+u{G}_{>}(0)/2]+O(u^2)$ and $u^{\prime }=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.

For a spherically symmetric gradient field regular at the origin and satisfying Poisson equation, the divergence theorem on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x{|}^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.

Let $U\subseteq D$ be open. Regard $H_{\mathrm{supp}}=H_0^1(U)$ as functions on $D$ by zero extension of H01, and let $H_{\mathrm{harm}}$ consist of the weakly harmonic Sobolev functions on $U$ belonging to $H_0^1(D)$. For $h\in H_{\mathrm{harm}}$, the Dirichlet inner product $(h,\varphi {)}_{\nabla }$ vanishes for each test function supported in $U$. Approximate any $u\in H_{\mathrm{supp}}$ by those test functions in the gradient norm and use the Cauchy-Schwarz inequality. This gives $\boxed{(h,u{)}_{\nabla }=0}$. The proof also works with the inhomogeneous zero-boundary Sobolev space convention, and with a homogeneous completion realized as weak functions.