Proof of A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the 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, $\langle Az,v\rangle_{H}=\langle z,v\rangle_{V}$ for $z\in D(A)$ and $v\in V$.

Since $L$ is Lipschitz with constant $\ell$, $|L(z)-L(v)|\le\ell|z-v|$ for $z,v\in H$, and taking $v=0_{H}$ 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, by Elementary Identities in a Real Inner Product Space §expansion, $\tfrac{1}{2}|c|^{2}+\langle u,c\rangle=\tfrac{1}{2}|c+u|^{2}-\tfrac{1}{2}|u|^{2}$ for $u,c\in H$. Secondly, for real $c>0$ and $u,v$ with $0\le u$ and $0\le v$, expanding $0\le\tfrac{1}{c}(c\,u-v)^{2}$ gives $2uv\le c\,u^{2}+\tfrac{v^{2}}{c}$; we use it as $uv\le\tfrac{1}{4}u^{2}+v^{2}$ (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 $H$. The defining expression does not contain its fourth argument, so $F(x,r,p,X)=F(x,r,p,X')$ for all $X,X'\in\mathrm{Sym}(V)$, which is First-Order Equation Operator on a Hilbert Triple §first-order. Degenerate ellipticity is then A First-Order Equation Operator is Degenerate Elliptic and Its $\delta$-Shifts Ignore the Form Argument §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 and claim 1, the $\delta$-shifts do not depend on their fourth argument and are given by

By the first elementary fact, with $c=p+\delta a$ and $u=a$, and with $c=p'-\delta b$ and $u=b$,

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 $K\alpha$ is nonnegative, being a product of positive numbers, so $t\mapsto K\alpha t$ 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 $X,Y\in\mathrm{Sym}(H)$ and let $\alpha,\delta\in\mathbb{R}$ with $1<\alpha$ and $0<\delta<1$. Put $w=\alpha(x-y)$ and $\sigma=\alpha|x-y|^{2}+\tfrac{1}{\alpha}$, and let $D=F^{-}_{\delta}(x,r,w,X)-F^{+}_{\delta}(y,r,w,Y)$. Expanding the two shift formulas and Elementary Identities in a Real Inner Product Space §expansion,

We bound the eight groups in turn, taking the fourth and the eighth together in (vii).

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

(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,

(iv) Since $0<\delta<1$ we have $\tfrac{1}{2}\le1-\tfrac{\delta}{2}$ and $\tfrac{1}{2}\le1+\tfrac{\delta}{2}$, 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 $0\le\omega_{g}^{\ast}(\sigma)$, $\omega_{g}^{\ast}$ 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 $-\omega_{g}^{\ast}(\sigma)$.

Suppose instead that $\alpha t^{2}<2C_{g}$. Multiplying by the positive $\tfrac{1}{\alpha}$ gives $t^{2}<2C_{g}\tfrac{1}{\alpha}$, while $\tfrac{1}{\alpha}\le\sigma$, the difference being the nonnegative $\alpha|x-y|^{2}$, and $0\le2C_{g}$ give $2C_{g}\tfrac{1}{\alpha}\le2C_{g}\sigma$ by claim 5 of Elementary Arithmetic in an Ordered Field. Hence $t^{2}\le2C_{g}\sigma$, and so $\bar{\omega}_{g}(t)\le\omega_{g}^{\ast}(\sigma)$ 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), (iii), (iv), (v), (vi) and (vii) account for all eight groups of $D$, the last of them for the fourth and the eighth 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 right-hand side is at least $-\omega_{1}(\sigma)-\omega_{2}(\delta(h(x)+h(y)+1),\alpha)$. This is the inequality required by The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple §pair, so $(\omega_{1},\omega_{2})$ is a structure pair for $F$ at $R$; as $R$ was an arbitrary positive real, $F$ satisfies the first-order structure condition by The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple §structure.

Claim 4, 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. If $(z,r,p,X)\in\mathcal{W}$ is $R$-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 and trichotomy, and $|z|\le\rho$. Let $\beta$ be a bound for $B$ on the set of $v\in V$ with $|v|_{V}\le\rho$, as provided by Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §bounded, put $\mu=c_{L}+\ell\rho$ and $N=\beta+\mu$; 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 and the bound on $|L|$.

Let $\xi=(x,r,p,X)$ be $R$-bounded. From $(*)$, using $\tfrac{1}{2}|p|^{2}\ge0$, The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|p|<R$, $1+\delta\le2$, the second elementary fact in the form $\langle L(x),a\rangle\ge-\tfrac{1}{4}|a|^{2}-|L(x)|^{2}$, Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §a-monotone, $0\le h(x)$, $-R\le r$ and $|g(x)|\le C_{g}$,

where $C_{-}=NR+\mu^{2}+\lambda_{0}R+C_{g}$; here $\tfrac{(1+\delta)^{2}-1}{2}=\delta+\tfrac{\delta^{2}}{2}\ge\delta$. Similarly, let $\eta=(y,s,p',X')$ be $R$-bounded. From $(**)$, using $\tfrac{1}{2}|p'|^{2}\le\tfrac{1}{2}R^{2}$, The Cauchy-Schwarz Inequality in a Real Inner Product Space, $1-\delta\le1$, $|\Phi(y)|\le N$, $0\le h(y)$, $s\le R$ and $-g(y)\le C_{g}$,

where $C_{+}=\tfrac{1}{2}R^{2}+NR+\lambda_{0}R+C_{g}$; here $\tfrac{1-(1-\delta)^{2}}{2}=\delta-\tfrac{\delta^{2}}{2}\ge\tfrac{\delta}{2}$ because $\delta<1$.

By the second elementary fact, $2R|a|\le\tfrac{3\delta}{8}|a|^{2}+\tfrac{8R^{2}}{3\delta}$ and $(R+N)|b|\le\tfrac{\delta}{4}|b|^{2}+\tfrac{(R+N)^{2}}{\delta}$. Hence the two displays give

where $E=\tfrac{8R^{2}}{3\delta}+\tfrac{(R+N)^{2}}{\delta}+C_{-}+C_{+}$; 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 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. 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 and trichotomy.

Claim 4, the shift modulus. Put $c=R+2\Lambda+N$ 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, and $s^{2}\le t^{2}$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, whence $\tfrac{1}{2}s^{2}\le\tfrac{1}{2}t^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field; 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. Since $F$, and hence each of its shifts, does not depend on its fourth argument, Elementary Identities in a Real Inner Product Space §expansion and The Cauchy-Schwarz Inequality in a Real Inner Product Space give

because $|P|+|a|+|\Phi(x)|\le(R+\Lambda)+\Lambda+N=c$. As $\omega$ is nondecreasing and $0\le\lVert Y\rVert$, we get $\omega(|q|)\le\omega(|q|+\lVert Y\rVert)$, which is the first requirement of The Shift-Continuity Condition on Admissible Test Data §modulus. Likewise, for $(y,s,p',X')\in S^{+}_{\delta,R}$, putting $P'=p'-\delta b$,

where $\tfrac{1}{2}|q|^{2}$ was discarded as nonnegative. 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.

Claim 5. By claims 1, 2, 3 and 4 the operator $F$ is a first-order second-order equation operator on $H$ that is locally strictly proper and satisfies the first-order structure condition and the shift-continuity condition, which are the hypotheses of A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition §comparison; applying that theorem to $u$, $v$ and $C$ gives $u(x)\le v(x)$ for every $x\in V$.