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 hypotheses $0\le\nu$, $0\le\theta$ and $\theta\le1$ are 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

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.

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

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

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

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,

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 $K\alpha+\nu\,\sigma(f)$ is nonnegative, being the sum of a product of positive numbers and a product of two nonnegative numbers, so $t\mapsto(K\alpha+\nu\,\sigma(f))\,t$ 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 Elementary Identities in a Real Inner Product Space §expansion on the two squared norms, whose terms $\tfrac{\theta}{2}|w|^{2}$ cancel, and using the two trace identities recorded there,

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,

(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,

(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 eighth group in (vii), where it absorbs the oscillation of $g$.

(iii) Likewise $\langle w,a+b\rangle=\alpha\bigl(\langle x,x-y\rangle_{V}+\langle y,x-y\rangle_{V}\bigr)=\alpha\bigl(|x|_{V}^{2}-|y|_{V}^{2}\bigr)=2\alpha\bigl(h(x)-h(y)\bigr)$, so, $h$ being nonnegative and $0\le\theta\le1$,

(iv) Since $0<\delta<1$ and $0\le\theta\le1$ we have $0\le\theta\delta\le1$, whence $\tfrac{1}{2}\le1-\tfrac{\theta\delta}{2}$ and $\tfrac{1}{2}\le1+\tfrac{\theta\delta}{2}$, so

(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,

(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

and likewise for $y$. Hence

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 fourth and the eighth 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

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 $\alpha t^{2}$,

In either case

(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 $\tfrac{\delta}{2}(|a|^{2}+|b|^{2})$ of (iv) and $-\tfrac{\delta}{4}(|a|^{2}+|b|^{2})$ of (vi) combine to the nonnegative $\tfrac{\delta}{4}(|a|^{2}+|b|^{2})$, and we obtain

Since $1<\alpha$ and $4c_{L}^{2}+4\ell^{2}$ is nonnegative, $2\alpha+4c_{L}^{2}+4\ell^{2}\le2\alpha+(4c_{L}^{2}+4\ell^{2})\alpha=K\alpha$, so the coefficient above is at most $K\alpha+\nu\,\sigma(f)$ 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

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

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 .

-\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}.

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_{-},

\tfrac{\theta}{2}|p'-\delta b|^{2}\ \le\ \tfrac{1}{2}R^{2}+\delta R|b|+\tfrac{\delta^{2}}{2}|b|^{2},

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

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

F^{-}{\delta}(x,r,p+q,X+Y)-F^{-}{\delta}(x,r,p,X)=-\tfrac{\nu}{2},\mathrm{Tr}{f}(Y|{V})+\theta\langle P,q\rangle+\tfrac{\theta}{2}|q|^{2}+\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},

F^{+}{\delta}(y,s,p'+q,X'+Y)-F^{+}{\delta}(y,s,p',X')=-\tfrac{\nu}{2},\mathrm{Tr}{f}(Y|{V})+\theta\langle P',q\rangle+\tfrac{\theta}{2}|q|^{2}+\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),