Proof of A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses

Each result cited is universally quantified over the data in its own statement. Throughout, $x,y\in D(A)$, and we write $a=Ax$, $b=Ay$ and $\Phi(z)=B(z)+L(z)$ for $z\in V$; norms and pairings without a subscript are those of $H$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §nonneg we have $0\le h(z)$ for $z\in V$, and $2h(z)=|z|_{V}^{2}$ by Hilbert Triples: Standing Notation and Background §penalty. By Hilbert Triples: Standing Notation and Background §triple we have $|z|_{H}\le|z|_{V}$ for $z\in V$. By Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator §symmetric, $\langle Az,\zeta\rangle_{H}=\langle z,\zeta\rangle_{V}$ for $z\in D(A)$ and $\zeta\in V$. For $Y\in\mathrm{Sym}(H)$ we write $Y|_{V}$ for its restriction to $V$. The standing hypothesis $0\le\nu$ is used throughout, as is $0\le\sigma(f)$, which holds by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §square-summable; the trace $\mathrm{Tr}_{f}$ is used only through Elementary Properties of the Trace of a Form along a Square-Summable Sequence.

Since $L$ is Lipschitz with constant $\ell$, $|L(z)-L(\zeta)|\le\ell|z-\zeta|$ for $z,\zeta\in H$, and taking $\zeta=0_{H}$ and using The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives

$$|L(z)|\le c_{L}+\ell|z|\qquad\text{for every }z\in H .$$

We use the following three elementary facts. First, Elementary Identities in a Real Inner Product Space §expansion gives $|c+u|^{2}=|c|^{2}+2\langle c,u\rangle+|u|^{2}$ for $c,u\in H$. Secondly, for real $c>0$ and $s,t$ with $0\le s$ and $0\le t$, expanding $0\le\tfrac{1}{c}(c\,s-t)^{2}$ gives $2st\le c\,s^{2}+\tfrac{t^{2}}{c}$; we use it as $st\le\tfrac{1}{4}s^{2}+t^{2}$ (the case $c=\tfrac{1}{2}$) and in the displayed forms below. Thirdly, $(s+t)^{2}\le2s^{2}+2t^{2}$ for real $s,t$, by Products and Sums of Weighted Square-Summable Sequences of Real Numbers §pointwise.

We also use three facts about $\Gamma$: for all $p,q\in H$ and $c\in\mathbb{R}$,

$$0\le\Gamma(p,p)\le|p|^{2},\qquad |\Gamma(p,q)|\le|p|\,|q|,\qquad \Gamma(p+cq,p+cq)=\Gamma(p,p)+2c\,\Gamma(p,q)+c^{2}\,\Gamma(q,q).$$

The first is the hypothesis $0_{\mathrm{Sym}}\preceq\Gamma\preceq I_{H}$ read through Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §order, since $0_{\mathrm{Sym}}(p,p)=0$ and $I_{H}(p,p)=\langle p,p\rangle=|p|^{2}$ by Hilbert Triples: Standing Notation and Background §restriction and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity. For the second, $-I_{H}\preceq0_{\mathrm{Sym}}$ because $-|p|^{2}\le0$ for every $p\in H$, so $-I_{H}\preceq\Gamma\preceq I_{H}$ by the transitivity in Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §order-partial; hence $\lVert\Gamma\rVert\le1$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §order-norm, and $|\Gamma(p,q)|\le\lVert\Gamma\rVert\,|p|\,|q|\le|p|\,|q|$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound. The third follows from the symmetry, additivity and homogeneity of $\Gamma$ in Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form.

Claim 1. The values of $F$ are real numbers and its domain is $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$, which with $W=D(A)$ is what Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §operator requires of a second-order equation operator on $H$. For degenerate ellipticity, let $x\in D(A)$, $r\in\mathbb{R}$, $p\in H$ and $X,Y\in\mathrm{Sym}(V)$ with $X\preceq Y$. Every term of the defining expression other than $-\tfrac{\nu}{2}\mathrm{Tr}_{f}X$ is independent of the fourth argument, and $\mathrm{Tr}_{f}X\le\mathrm{Tr}_{f}Y$ by Elementary Properties of the Trace of a Form along a Square-Summable Sequence §monotone, so

$$F(x,r,p,Y)-F(x,r,p,X)=-\tfrac{\nu}{2}\bigl(\mathrm{Tr}_{f}Y-\mathrm{Tr}_{f}X\bigr)\le0,$$

because $\tfrac{\nu}{2}$ and $\mathrm{Tr}_{f}Y-\mathrm{Tr}_{f}X$ are nonnegative, by claim 5 of Elementary Arithmetic in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field. Thus $F(x,r,p,Y)\le F(x,r,p,X)$, which is Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple §elliptic.

Claim 2. Let $R$ be positive, $x\in D(A)$, $p\in H$, $X\in\mathrm{Sym}(V)$ and $r,s\in\mathbb{R}$ with $-R\le s\le r\le R$. Every term of $F$ other than $\lambda_{0}r$ is independent of the second argument, so

$$F(x,r,p,X)-F(x,s,p,X)=\lambda_{0}r-\lambda_{0}s=\lambda_{0}(r-s),$$

and in particular $\lambda_{0}(r-s)\le F(x,r,p,X)-F(x,s,p,X)$. As $\lambda_{0}$ is positive, this is the requirement of Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple §constant; since $R$ was an arbitrary positive real, $F$ is locally strictly proper by Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple §strictly-proper.

The shifts. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$. By Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted, the $\delta$-shifts are given by

$$F^{-}_{\delta}(x,r,p,X)=\lambda_{0}\bigl(r+\delta h(x)\bigr)-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}\bigl(X|_{V}+\delta I_{V}\bigr)+\tfrac12\,\Gamma(p+\delta a,p+\delta a)+\langle a+\Phi(x),p+\delta a\rangle-g(x),$$ $$F^{+}_{\delta}(y,s,p',X')=\lambda_{0}\bigl(s-\delta h(y)\bigr)-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}\bigl(X'|_{V}-\delta I_{V}\bigr)+\tfrac12\,\Gamma(p'-\delta b,p'-\delta b)+\langle b+\Phi(y),p'-\delta b\rangle-g(y),$$

for $x,y\in D(A)$, $r,s\in\mathbb{R}$, $p,p'\in H$ and $X,X'\in\mathrm{Sym}(H)$. By Elementary Properties of the Trace of a Form along a Square-Summable Sequence §linear and Elementary Properties of the Trace of a Form along a Square-Summable Sequence §identity,

$$\mathrm{Tr}_{f}\bigl(X|_{V}+\delta I_{V}\bigr)=\mathrm{Tr}_{f}(X|_{V})+\delta\,\sigma(f), \qquad \mathrm{Tr}_{f}\bigl(X'|_{V}-\delta I_{V}\bigr)=\mathrm{Tr}_{f}(X'|_{V})-\delta\,\sigma(f).$$

Claim 3, the two moduli. By The Nondecreasing Envelope of a Truncated Modulus of Continuity §modulus the function $\bar{\omega}_{g}$ is a modulus of continuity satisfying $0\le\bar{\omega}_{g}(t)\le2C_{g}$ for every nonnegative $t$; as $2C_{g}$ is nonnegative, its quadratic reparametrisation $\omega_{g}^{\ast}$ at $2C_{g}$ is therefore defined, and is a nondecreasing modulus of continuity, by Sums, Nonnegative Multiples, Monotonicity and Quadratic Reparametrisation of Moduli of Continuity §quadratic. By Linear Moduli of Continuity §modulus the function $t\mapsto\ell t$ is a modulus of continuity, since $0\le\ell$; so $\omega_{1}$ is a modulus of continuity by Sums, Nonnegative Multiples, Monotonicity and Quadratic Reparametrisation of Moduli of Continuity §sum. For a real $\alpha>1$ the number $K\alpha^{2}+\nu\,\sigma(f)$ is nonnegative, being the sum of a product of positive numbers and a product of two nonnegative numbers, so $t\mapsto(K\alpha^{2}+\nu\,\sigma(f))\,t$ is a modulus of continuity by Linear Moduli of Continuity §modulus.

Claim 3, the inequality. Let $R$ be positive, let $x,y\in D(A)$, let $r\in\mathbb{R}$ with $-R\le r\le R$, let $\alpha,\delta\in\mathbb{R}$ with $1<\alpha$ and $0<\delta<1$, and let $(X,Y)$ be a pair of members of $\mathrm{Sym}(H)$ admitted at $\alpha$. Put $w=\alpha(x-y)$ and $\tau=\alpha|x-y|^{2}+\tfrac{1}{\alpha}$, and let $D=F^{-}_{\delta}(x,r,w,X)-F^{+}_{\delta}(y,r,w,Y)$. Expanding the two shift formulas of the previous paragraph, using the third fact about $\Gamma$ on the two quadratic terms in the gradient, whose terms $\tfrac12\Gamma(w,w)$ cancel, and using the two trace identities recorded there,

$$D=\lambda_{0}\delta\bigl(h(x)+h(y)\bigr)+\tfrac{\nu}{2}\bigl(\mathrm{Tr}_{f}(Y|_{V})-\mathrm{Tr}_{f}(X|_{V})\bigr)-\nu\delta\,\sigma(f)+\delta\,\Gamma(w,a+b)+\tfrac{\delta^{2}}{2}\bigl(\Gamma(a,a)-\Gamma(b,b)\bigr)+\langle a-b,w\rangle+\langle\Phi(x)-\Phi(y),w\rangle+\delta\bigl(|a|^{2}+|b|^{2}\bigr)+\delta\bigl(\langle\Phi(x),a\rangle+\langle\Phi(y),b\rangle\bigr)-\bigl(g(x)-g(y)\bigr).$$

We bound the ten groups in turn, taking the sixth and the tenth together in (vii).

(i) $0\le\lambda_{0}\delta(h(x)+h(y))$, all three factors being nonnegative.

(i-a) Being admitted at $\alpha$, the pair satisfies $X\preceq Y$; hence $X|_{V}\preceq Y|_{V}$ by claim 10 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity and therefore $\mathrm{Tr}_{f}(X|_{V})\le\mathrm{Tr}_{f}(Y|_{V})$ by Elementary Properties of the Trace of a Form along a Square-Summable Sequence §monotone. As $\tfrac{\nu}{2}$ is nonnegative,

$$0\le\tfrac{\nu}{2}\bigl(\mathrm{Tr}_{f}(Y|_{V})-\mathrm{Tr}_{f}(X|_{V})\bigr).$$

(i-b) Since $0\le h(x)+h(y)$ we have $\delta\le\delta(h(x)+h(y)+1)$, and $\nu\,\sigma(f)$ is nonnegative, so by claim 5 of Elementary Arithmetic in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field,

$$-\nu\delta\,\sigma(f)\ \ge\ -\nu\,\sigma(f)\cdot\delta\bigl(h(x)+h(y)+1\bigr).$$

(ii) Since $x-y\in V$, the symmetry identity gives $\langle a-b,w\rangle=\alpha\bigl(\langle x,x-y\rangle_{V}-\langle y,x-y\rangle_{V}\bigr)=\alpha|x-y|_{V}^{2}$, which is nonnegative. This term is not discarded: it is combined with the tenth group in (vii), where it absorbs the oscillation of $g$.

(iii) By the second fact about $\Gamma$ and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle, $\delta\,\Gamma(w,a+b)\ge-\delta|w|\,|a+b|\ge-\delta|w|\,|a|-\delta|w|\,|b|$, and the second elementary fact with $c=\tfrac{1}{16}$ gives $|w|\,|a|\le\tfrac{1}{32}|a|^{2}+8|w|^{2}$ and $|w|\,|b|\le\tfrac{1}{32}|b|^{2}+8|w|^{2}$. By Elementary Identities in a Real Inner Product Space §homogeneity and $0<\alpha$, by Elementary Identities in a Real Inner Product Space §parallelogram together with $0\le|x+y|^{2}$, and by $|x|\le|x|_{V}$, $|y|\le|y|_{V}$ with claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field,

$$|w|^{2}=\alpha^{2}|x-y|^{2}\le\alpha^{2}\bigl(2|x|^{2}+2|y|^{2}\bigr)\le2\alpha^{2}\bigl(|x|_{V}^{2}+|y|_{V}^{2}\bigr)=4\alpha^{2}\bigl(h(x)+h(y)\bigr).$$

Hence, as $h(x)+h(y)\le h(x)+h(y)+1$ and the coefficients are nonnegative,

$$\delta\,\Gamma(w,a+b)\ \ge\ -\tfrac{\delta}{32}\bigl(|a|^{2}+|b|^{2}\bigr)-16\delta|w|^{2}\ \ge\ -\tfrac{\delta}{32}\bigl(|a|^{2}+|b|^{2}\bigr)-64\alpha^{2}\cdot\delta\bigl(h(x)+h(y)+1\bigr).$$

(iv) By the first fact about $\Gamma$, $0\le\Gamma(a,a)$ and $\Gamma(b,b)\le|b|^{2}$, and $\delta^{2}\le\delta$ since $0<\delta<1$; so

$$\tfrac{\delta^{2}}{2}\bigl(\Gamma(a,a)-\Gamma(b,b)\bigr)+\delta\bigl(|a|^{2}+|b|^{2}\bigr)\ \ge\ \delta|a|^{2}+\bigl(\delta-\tfrac{\delta^{2}}{2}\bigr)|b|^{2}\ \ge\ \tfrac{\delta}{2}\bigl(|a|^{2}+|b|^{2}\bigr).$$

(v) $\langle\Phi(x)-\Phi(y),w\rangle=\alpha\langle B(x)-B(y),x-y\rangle+\alpha\langle L(x)-L(y),x-y\rangle$. The first summand is nonnegative by Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §monotone and $0<\alpha$. For the second, The Cauchy-Schwarz Inequality in a Real Inner Product Space and the Lipschitz bound give $\langle L(x)-L(y),x-y\rangle\ge-\ell|x-y|^{2}$, so, $\tfrac{1}{\alpha}$ being positive and $\ell$ nonnegative,

$$\langle\Phi(x)-\Phi(y),w\rangle\ge-\ell\,\alpha|x-y|^{2}\ge-\ell\,\tau .$$

(vi) $\delta(\langle\Phi(x),a\rangle+\langle\Phi(y),b\rangle)=\delta(\langle B(x),a\rangle+\langle B(y),b\rangle)+\delta(\langle L(x),a\rangle+\langle L(y),b\rangle)$. The first summand is nonnegative by Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §a-monotone. For the second, The Cauchy-Schwarz Inequality in a Real Inner Product Space and the second elementary fact give $\langle L(x),a\rangle\ge-|L(x)|\,|a|\ge-\tfrac{1}{4}|a|^{2}-|L(x)|^{2}$, while the third elementary fact, the Lipschitz bound and $|x|\le|x|_{V}$ give

$$|L(x)|^{2}\le\bigl(c_{L}+\ell|x|\bigr)^{2}\le2c_{L}^{2}+2\ell^{2}|x|^{2}\le2c_{L}^{2}+2\ell^{2}|x|_{V}^{2}=2c_{L}^{2}+4\ell^{2}h(x),$$

and likewise for $y$. Hence

$$\delta\bigl(\langle\Phi(x),a\rangle+\langle\Phi(y),b\rangle\bigr)\ \ge\ -\tfrac{\delta}{4}\bigl(|a|^{2}+|b|^{2}\bigr)-\delta\Bigl(4c_{L}^{2}+4\ell^{2}\bigl(h(x)+h(y)\bigr)\Bigr)\ \ge\ -\tfrac{\delta}{4}\bigl(|a|^{2}+|b|^{2}\bigr)-\bigl(4c_{L}^{2}+4\ell^{2}\bigr)\,\delta\bigl(h(x)+h(y)+1\bigr),$$

the last step because $1\le h(x)+h(y)+1$ and $h(x)+h(y)\le h(x)+h(y)+1$ and the coefficients are nonnegative.

(vii) We bound the sixth and the tenth group together. Write $t=|x-y|_{V}$, so that their sum is $\alpha t^{2}-\bigl(g(x)-g(y)\bigr)$ by (ii). Since $x,y\in D(A)\subseteq V$, the hypotheses on $g$ give $g(x)-g(y)\le|g(x)-g(y)|\le\omega_{g}(t)$, by claim 3 of Properties of the Absolute Value in an Ordered Field, and $|g(x)-g(y)|\le|g(x)|+|g(y)|\le2C_{g}$, by claims 2 and 5 of that lemma; so $g(x)-g(y)$ is at most both $\omega_{g}(t)$ and $2C_{g}$ and therefore

$$g(x)-g(y)\le\bar{\omega}_{g}(t)$$

by The Nondecreasing Envelope of a Truncated Modulus of Continuity §majorant. Note that $\alpha t^{2}$ is nonnegative, being a product of the positive $\alpha$ with a square, and that $0\le\omega_{g}^{\ast}(\tau)$, $\omega_{g}^{\ast}$ being a modulus of continuity.

Suppose first that $2C_{g}\le\alpha t^{2}$. Since $\bar{\omega}_{g}(t)\le2C_{g}$ by The Nondecreasing Envelope of a Truncated Modulus of Continuity §modulus, transitivity gives $g(x)-g(y)\le\alpha t^{2}$, so $\alpha t^{2}-(g(x)-g(y))$ is nonnegative and hence at least $-\omega_{g}^{\ast}(\tau)$.

Suppose instead that $\alpha t^{2}<2C_{g}$. Multiplying by the positive $\tfrac{1}{\alpha}$ gives $t^{2}<2C_{g}\tfrac{1}{\alpha}$, while $\tfrac{1}{\alpha}\le\tau$, the difference being the nonnegative $\alpha|x-y|^{2}$, and $0\le2C_{g}$ give $2C_{g}\tfrac{1}{\alpha}\le2C_{g}\tau$ by claim 5 of Elementary Arithmetic in an Ordered Field. Hence $t^{2}\le2C_{g}\tau$, and so $\bar{\omega}_{g}(t)\le\omega_{g}^{\ast}(\tau)$ by Sums, Nonnegative Multiples, Monotonicity and Quadratic Reparametrisation of Moduli of Continuity §quadratic, the reparametrisation being taken at $c=2C_{g}$. Discarding the nonnegative $\alpha t^{2}$,

$$\alpha t^{2}-\bigl(g(x)-g(y)\bigr)\ \ge\ -\bar{\omega}_{g}(t)\ \ge\ -\omega_{g}^{\ast}(\tau).$$

In either case

$$\langle a-b,w\rangle-\bigl(g(x)-g(y)\bigr)\ \ge\ -\omega_{g}^{\ast}(\tau).$$

(viii) The bounds (i), (i-a), (i-b), (iii), (iv), (v), (vi) and (vii) account for all ten groups of $D$, the last of them for the sixth and the tenth together. Adding them, the terms $-\tfrac{\delta}{32}(|a|^{2}+|b|^{2})$ of (iii), $\tfrac{\delta}{2}(|a|^{2}+|b|^{2})$ of (iv) and $-\tfrac{\delta}{4}(|a|^{2}+|b|^{2})$ of (vi) combine to the nonnegative $\tfrac{7\delta}{32}(|a|^{2}+|b|^{2})$, and we obtain

$$D\ \ge\ -\omega_{g}^{\ast}(\tau)-\ell\,\tau-\bigl(64\alpha^{2}+4c_{L}^{2}+4\ell^{2}+\nu\,\sigma(f)\bigr)\,\delta\bigl(h(x)+h(y)+1\bigr).$$

Since $1<\alpha$ we have $1<\alpha^{2}$, and as $4c_{L}^{2}+4\ell^{2}$ is nonnegative, $64\alpha^{2}+4c_{L}^{2}+4\ell^{2}\le64\alpha^{2}+(4c_{L}^{2}+4\ell^{2})\alpha^{2}=K\alpha^{2}$, so the coefficient above is at most $K\alpha^{2}+\nu\,\sigma(f)$ and the right-hand side is at least $-\omega_{1}(\tau)-\omega_{2}(\delta(h(x)+h(y)+1),\alpha)$. This is the inequality required by The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple §pair, whose displayed hypothesis is imposed only for pairs admitted at $\alpha$, as assumed here; so $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$, and as $R$ was an arbitrary positive real, $F$ satisfies the second-order structure condition by The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple §structure.

Claim 4. Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$, and let $(N_{m})_{m\in\mathbb{N}}$ be its sequence of tail forms, as in The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §tail-forms. Let $\beta,R,\rho\in\mathbb{R}$ be positive and let $\delta\in\mathbb{R}$ satisfy $0<\delta<1$.

By Elementary Properties of the Trace of a Form along a Square-Summable Sequence §tail the sequence $(\mathrm{Tr}_{f}(N_{m}|_{V}))_{m\in\mathbb{N}}$ converges to $0$, so by claim 3 of Arithmetic of Limits of Real Sequences the sequence with $m$-th term $\tfrac{\nu\beta}{2}\mathrm{Tr}_{f}(N_{m}|_{V})$ converges to $0$ as well; since its terms are nonnegative, $\tfrac{\nu\beta}{2}$ and $\mathrm{Tr}_{f}(N_{m}|_{V})$ being so, there is $m_{0}\in\mathbb{N}$ with

$$\tfrac{\nu\beta}{2}\,\mathrm{Tr}_{f}\bigl(N_{m}|_{V}\bigr)\le\rho\qquad\text{for every }m\in\mathbb{N}\text{ with }m_{0}\le m .$$

Let $m\in\mathbb{N}$ satisfy $m_{0}\le m$, and let $x\in D(A)$, $r\in\mathbb{R}$, $p\in H$ and $X\in\mathrm{Sym}(H)$. By claim 10 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity we have $(X+\beta N_{m})|_{V}=X|_{V}+\beta\,N_{m}|_{V}$ and $(X-\beta N_{m})|_{V}=X|_{V}-\beta\,N_{m}|_{V}$, so Elementary Properties of the Trace of a Form along a Square-Summable Sequence §linear gives

\mathrm{Tr}_{f}\bigl((X+\beta N_{m})|_{V}\bigr)=\mathrm{Tr}_{f}(X|_{V})+\beta\,\mathrm{Tr}_{f}\bigl(N_{m}|_{V}\bigr), \qquad \mathrm{Tr}_{f}\bigl((X-\beta N_{m})|_{V}\bigr)=\mathrm{Tr}_{f}(X|_{V})-\beta\,\mathrm{Tr}_{f}\bigl(N_{m}|_{V}\bigr). $$ Every term of the two shifts other than the trace term is independent of the fourth argument, so, using the shift formulas and the trace identities recorded above,

F^{-}{\delta}(x,r,p,X)-F^{-}{\delta}\bigl(x,r,p,X+\beta N_{m}\bigr)=\tfrac{\nu\beta}{2},\mathrm{Tr}{f}\bigl(N{m}|_{V}\bigr)\le\rho,

$$$$

F^{+}{\delta}\bigl(x,r,p,X-\beta N{m}\bigr)-F^{+}{\delta}(x,r,p,X)=\tfrac{\nu\beta}{2},\mathrm{Tr}{f}\bigl(N_{m}|_{V}\bigr)\le\rho .

These hold for all such data, in particular for the terms of any sequences as in [The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §lower](/theorems/7a1fad52-e5e9-4b06-b9f7-3404ec390a28?v=60a64940-3e71-4244-88de-b6da3ae47192#clause-lower) and [The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §upper](/theorems/7a1fad52-e5e9-4b06-b9f7-3404ec390a28?v=60a64940-3e71-4244-88de-b6da3ae47192#clause-upper), whose $R$-boundedness hypothesis is not needed; rearranged, they are the two inequalities required there. Hence $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$. If moreover $H$ is not finite-dimensional as a vector space over $\mathbb{R}$, such a basis exists by [An Orthonormal Basis of the Ambient Space Contained in the Form Space of a Hilbert Triple §basis](/theorems/708f4661-d0e5-4c89-84a3-f4ac1b404da7?v=f8b276c1-f241-4fc2-873e-fe11923c2357#clause-basis), and $F$ then satisfies the tail-insensitivity condition by [The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §condition](/theorems/7a1fad52-e5e9-4b06-b9f7-3404ec390a28?v=60a64940-3e71-4244-88de-b6da3ae47192#clause-condition). **Claim 5, a bound on the form operator over the admissible sets.** Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$. Let $\rho$ be the nonnegative real with $\rho^{2}=2R$, given by [Existence and Uniqueness of the Nonnegative Square Root](/theorems/7f783996-7e13-4f8a-9353-21a467864e85?v=9fba8150-5ca0-4dd5-911f-aa0f5cf5feec). If $(z,r,p,X)\in\mathcal{W}$ is [$R$-bounded](/theorems/4640aebc-d86c-4e8f-8b51-dbba74a2d2be?v=5d4402f9-29f2-408b-9701-41c882b2060b#clause-bounded) then $h(z)<R$, so $|z|_{V}^{2}=2h(z)<\rho^{2}$ and therefore $|z|_{V}\le\rho$ by claim 1 of [Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field](/theorems/c4e16e3c-4449-4681-953c-02ad68f9842b?v=d3a024bb-6971-4912-9ff5-7923ca2a4893) and trichotomy, and $|z|\le\rho$. Let $\beta_{0}$ be a bound for $B$ on the set of $\zeta\in V$ with $|\zeta|_{V}\le\rho$, as provided by [Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §bounded](/theorems/ffd56497-3c4f-4c81-8443-c7c97b91b400?v=fae9a4a6-fde1-4289-920d-00ad75746aea#clause-bounded), put $\mu=c_{L}+\ell\rho$ and $N=\beta_{0}+\mu$; then $|\Phi(z)|\le N$ for every such $z$, by [The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle](/theorems/0c7d58a7-82ab-48c0-a8be-ad7b9e2297fc?v=237d76f6-5e7b-450e-974e-4bf327ff854e#clause-triangle) and the bound on $|L|$. We first bound the trace terms. If $(z,r,p,X)$ is $R$-bounded then $\lVert X\rVert<R$, so [Elementary Properties of the Trace of a Form along a Square-Summable Sequence §restriction](/theorems/d74946b5-13a2-4657-81e2-bf67736f3aa1?v=d0d0bd01-a849-4393-b600-07e56979fdaf#clause-restriction) gives $|\mathrm{Tr}_{f}(X|_{V})|\le R\,\sigma(f)$, while $\delta<1$ and $0\le\sigma(f)$ give $\delta\,\sigma(f)\le\sigma(f)$; with the trace identities recorded above and $0\le\tfrac{\nu}{2}$ this yields, putting $C_{\nu}=\tfrac{\nu}{2}(R+1)\,\sigma(f)$,

-\tfrac{\nu}{2},\mathrm{Tr}{f}\bigl(X|{V}+\delta I_{V}\bigr)\ \ge\ -C_{\nu}, \qquad -\tfrac{\nu}{2},\mathrm{Tr}{f}\bigl(X|{V}-\delta I_{V}\bigr)\ \le\ C_{\nu}.

Let $\xi=(x,r,p,X)$ be $R$-bounded. From the formula for $F^{-}_{\delta}$, using $0\le\tfrac12\Gamma(p+\delta a,p+\delta a)$ by the first fact about $\Gamma$, [The Cauchy-Schwarz Inequality in a Real Inner Product Space](/theorems/75e68dc9-c425-4d4e-afd8-53eba404a50e?v=638ad390-44c9-472a-bccb-9ba6242bb219) with $|p|<R$, the second elementary fact in the form $\langle L(x),a\rangle\ge-\tfrac{1}{4}|a|^{2}-|L(x)|^{2}$ together with $|L(x)|\le\mu$ and $\delta<1$, [Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §a-monotone](/theorems/ffd56497-3c4f-4c81-8443-c7c97b91b400?v=fae9a4a6-fde1-4289-920d-00ad75746aea#clause-a-monotone), $|\Phi(x)|\le N$, $0\le h(x)$, $-R\le r$ and $|g(x)|\le C_{g}$,

F^{-}{\delta}(\xi)\ \ge\ \delta|a|^{2}-R|a|-NR-\tfrac{\delta}{4}|a|^{2}-\mu^{2}-\lambda{0}R-C_{g}-C_{\nu}\ =\ \tfrac{3\delta}{4}|a|^{2}-R|a|-C_{-},

where $C_{-}=NR+\mu^{2}+\lambda_{0}R+C_{g}+C_{\nu}$. Similarly, let $\eta=(y,s,p',X')$ be $R$-bounded. Bounding $\tfrac12\Gamma(p'-\delta b,p'-\delta b)$ by $\tfrac12|p'-\delta b|^{2}$, by the first fact about $\Gamma$, expanding the latter by the first elementary fact and using $|p'|<R$, [The Cauchy-Schwarz Inequality in a Real Inner Product Space](/theorems/75e68dc9-c425-4d4e-afd8-53eba404a50e?v=638ad390-44c9-472a-bccb-9ba6242bb219) and $0<\delta<1$,

\tfrac12\Gamma(p'-\delta b,p'-\delta b)\ \le\ \tfrac{1}{2}R^{2}+\delta R|b|+\tfrac{\delta^{2}}{2}|b|^{2},

while $\langle b+\Phi(y),p'-\delta b\rangle\le R|b|-\delta|b|^{2}+NR+\delta N|b|$ by [The Cauchy-Schwarz Inequality in a Real Inner Product Space](/theorems/75e68dc9-c425-4d4e-afd8-53eba404a50e?v=638ad390-44c9-472a-bccb-9ba6242bb219) and $|\Phi(y)|\le N$. Adding these, and using $s\le R$, $0\le h(y)$, $-g(y)\le C_{g}$, the trace bound above, $\delta<1$ and $\tfrac{\delta^{2}}{2}-\delta\le-\tfrac{\delta}{2}$,

F^{+}{\delta}(\eta)\ \le\ -\tfrac{\delta}{2}|b|^{2}+(2R+N)|b|+C{+},

where $C_{+}=\tfrac{1}{2}R^{2}+NR+\lambda_{0}R+C_{g}+C_{\nu}$. By the second elementary fact, $R|a|\le\tfrac{3\delta}{8}|a|^{2}+\tfrac{2R^{2}}{3\delta}$ and $(2R+N)|b|\le\tfrac{\delta}{4}|b|^{2}+\tfrac{(2R+N)^{2}}{\delta}$. Hence the two displays give

F^{-}{\delta}(\xi)\ \ge\ \tfrac{3\delta}{8}|a|^{2}-E,\qquad F^{+}{\delta}(\eta)\ \le\ -\tfrac{\delta}{4}|b|^{2}+E,

where $E=\tfrac{2R^{2}}{3\delta}+\tfrac{(2R+N)^{2}}{\delta}+C_{-}+C_{+}$; in particular $F^{-}_{\delta}(\xi)\ge-E$ and $F^{+}_{\delta}(\eta)\le E$ for all $R$-bounded $\xi$ and $\eta$. Now let $\xi=(x,r,p,X)\in S^{-}_{\delta,R}$. By [Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets §admissible](/theorems/4640aebc-d86c-4e8f-8b51-dbba74a2d2be?v=5d4402f9-29f2-408b-9701-41c882b2060b#clause-admissible) there is an $R$-bounded $\eta$ with $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R$, so $\tfrac{3\delta}{8}|a|^{2}-E\le F^{-}_{\delta}(\xi)<R+F^{+}_{\delta}(\eta)\le R+E$ and therefore $|a|^{2}<\tfrac{8(R+2E)}{3\delta}$. Similarly, if $\eta=(y,s,p',X')\in S^{+}_{\delta,R}$ there is an $R$-bounded $\xi$ with $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R$, so $-\tfrac{\delta}{4}|b|^{2}+E\ge F^{+}_{\delta}(\eta)>F^{-}_{\delta}(\xi)-R\ge-E-R$ and therefore $|b|^{2}<\tfrac{4(R+2E)}{\delta}$. Let $\Lambda$ be the nonnegative real whose square is the larger of $\tfrac{8(R+2E)}{3\delta}$ and $\tfrac{4(R+2E)}{\delta}$, again by [Existence and Uniqueness of the Nonnegative Square Root](/theorems/7f783996-7e13-4f8a-9353-21a467864e85?v=9fba8150-5ca0-4dd5-911f-aa0f5cf5feec). Then $|Ax|\le\Lambda$ whenever $(x,r,p,X)$ lies in $S^{-}_{\delta,R}$ or in $S^{+}_{\delta,R}$, by claim 1 of [Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field](/theorems/c4e16e3c-4449-4681-953c-02ad68f9842b?v=d3a024bb-6971-4912-9ff5-7923ca2a4893) and trichotomy. **Claim 5, the shift modulus.** Put $c=R+2\Lambda+N+\tfrac{\nu}{2}\,\sigma(f)$ and let $\omega$ be the function on the nonnegative reals with $\omega(t)=c\,t+\tfrac{1}{2}t^{2}$. It is nonnegative there. Given a positive $\varepsilon$, let $\delta_{0}$ be the lesser of $1$ and $\tfrac{\varepsilon}{c+1}$, which is positive; every nonnegative $t\le\delta_{0}$ satisfies $t\le1$, hence $t^{2}\le t$ and $\omega(t)\le(c+1)t\le\varepsilon$. So $\omega$ is a modulus of continuity. It is also nondecreasing on the nonnegative reals: for $0\le s\le t$ we have $c\,s\le c\,t$ by [Linear Moduli of Continuity §monotone](/theorems/206eec08-b1fd-4f5b-a4a1-42fc2a8ddbc8?v=be112d1c-eb90-48ca-9371-16fa3ebca6c7#clause-monotone), and $s^{2}\le t^{2}$ by claim 2 of [Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field](/theorems/c4e16e3c-4449-4681-953c-02ad68f9842b?v=d3a024bb-6971-4912-9ff5-7923ca2a4893), whence $\tfrac{1}{2}s^{2}\le\tfrac{1}{2}t^{2}$ by claim 5 of [Elementary Arithmetic in an Ordered Field](/theorems/fe1c552a-759a-469b-86a4-5207560aefaf?v=63c2af8b-b56f-40d4-bc16-3fb0690235bc); adding the two inequalities by claims 3 and 2 of that lemma gives $\omega(s)\le\omega(t)$. Let $q\in H$ and $Y\in\mathrm{Sym}(H)$. Let $(x,r,p,X)\in S^{-}_{\delta,R}$ and put $P=p+\delta a$, so that $|P|\le R+\delta\Lambda\le R+\Lambda$ by [The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle](/theorems/0c7d58a7-82ab-48c0-a8be-ad7b9e2297fc?v=237d76f6-5e7b-450e-974e-4bf327ff854e#clause-triangle). By the third fact about $\Gamma$, [Elementary Identities in a Real Inner Product Space §bilinear](/theorems/5ce8e666-53b7-4958-beed-be505dcf38ba?v=5bbf7092-58eb-4039-aad3-ddfbf22d02dd#clause-bilinear), [The Cauchy-Schwarz Inequality in a Real Inner Product Space](/theorems/75e68dc9-c425-4d4e-afd8-53eba404a50e?v=638ad390-44c9-472a-bccb-9ba6242bb219), [Elementary Properties of the Trace of a Form along a Square-Summable Sequence §linear](/theorems/d74946b5-13a2-4657-81e2-bf67736f3aa1?v=d0d0bd01-a849-4393-b600-07e56979fdaf#clause-linear) together with claim 10 of [Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity](/theorems/ee1a6501-eb70-41c9-a3ea-2490110e3a3e?v=6894704a-d362-4438-a83d-906109126b6b), and the first two facts about $\Gamma$, which give $\Gamma(P,q)\le|P|\,|q|$ and $\tfrac12\Gamma(q,q)\le\tfrac12|q|^{2}$,

F^{-}{\delta}(x,r,p+q,X+Y)-F^{-}{\delta}(x,r,p,X)=-\tfrac{\nu}{2},\mathrm{Tr}{f}(Y|{V})+\Gamma(P,q)+\tfrac12\Gamma(q,q)+\langle a+\Phi(x),q\rangle\ \le\ \tfrac{\nu}{2},\sigma(f),\lVert Y\rVert+\bigl(|P|+|a|+|\Phi(x)|\bigr)|q|+\tfrac{1}{2}|q|^{2},

the trace term being bounded by [Elementary Properties of the Trace of a Form along a Square-Summable Sequence §restriction](/theorems/d74946b5-13a2-4657-81e2-bf67736f3aa1?v=d0d0bd01-a849-4393-b600-07e56979fdaf#clause-restriction). Write $t=|q|+\lVert Y\rVert$, so that $|q|\le t$ and $\lVert Y\rVert\le t$, both being nonnegative; then $|q|^{2}\le t^{2}$ by claim 2 of [Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field](/theorems/c4e16e3c-4449-4681-953c-02ad68f9842b?v=d3a024bb-6971-4912-9ff5-7923ca2a4893), and $|P|+|a|+|\Phi(x)|\le(R+\Lambda)+\Lambda+N$, so, all the coefficients being nonnegative, the right-hand side is at most $c\,t+\tfrac{1}{2}t^{2}=\omega(|q|+\lVert Y\rVert)$. This is the first requirement of [The Shift-Continuity Condition on Admissible Test Data §modulus](/theorems/d5ee1a2e-5473-4025-b8b8-daa65131c780?v=2618888e-84e7-485a-b732-842dc4ebefe4#clause-modulus). Likewise, for $(y,s,p',X')\in S^{+}_{\delta,R}$, putting $P'=p'-\delta b$,

F^{+}{\delta}(y,s,p'+q,X'+Y)-F^{+}{\delta}(y,s,p',X')=-\tfrac{\nu}{2},\mathrm{Tr}{f}(Y|{V})+\Gamma(P',q)+\tfrac12\Gamma(q,q)+\langle b+\Phi(y),q\rangle\ \ge\ -\tfrac{\nu}{2},\sigma(f),\lVert Y\rVert-\bigl(|P'|+|b|+|\Phi(y)|\bigr)|q|\ \ge\ -\omega(|q|+\lVert Y\rVert),

where $\Gamma(P',q)\ge-|P'|\,|q|$ by the second fact about $\Gamma$, and $\tfrac12\Gamma(q,q)$ was discarded as nonnegative by the first. This is the second requirement. Hence $\omega$ is a shift modulus for $F$ at $(\delta,R)$, and as $\delta$ and $R$ were arbitrary, $F$ satisfies the shift-continuity condition by [The Shift-Continuity Condition on Admissible Test Data §continuity](/theorems/d5ee1a2e-5473-4025-b8b8-daa65131c780?v=2618888e-84e7-485a-b732-842dc4ebefe4#clause-continuity). **Claim 6.** Here $H$ is assumed not finite-dimensional. By claims 1, 2, 3, 4 and 5 the operator $F$ is a second-order equation operator on $H$ that is degenerate elliptic and locally strictly proper and satisfies the second-order structure condition, the tail-insensitivity condition and the shift-continuity condition, which are the hypotheses of [A Comparison Principle on a Hilbert Triple under the Second-Order Structure Condition §comparison](/theorems/39ba1c43-8484-4f3f-baee-579cdf02184e?v=877e17f8-b7fc-488b-a750-38cd723ba627#clause-comparison); applying that theorem to $u$, $v$ and $C$ gives $u(x)\le v(x)$ for every $x\in V$.