Proof of Well-Posedness of the Eikonal and Discounted Hopf-Lax Equations on Noncommutative Laws with Distance Data

Each result cited below is universally quantified over the data in its own statement.

Throughout, $X=\Sigma_{d,R}$ with the metric $W_{2}$, and $K$, $\Omega$ and $D$ are as in the statement. By The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §complete the metric space $(\Sigma_{d,R},W_{2})$ is complete, and by The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §interpolation it has interpolation points; $K$ is nonempty, and $\Omega=\Sigma_{d,R}\setminus K$ is open by Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance §closed. For reals $s,c$ we use claim 9 of Properties of the Absolute Value in an Ordered Field: $|s|<c$ if and only if $-c<s$ and $s<c$.

Step 1 (Bounds on $W_{2}$ and $D$). Put $B=1+4dR^{2}$, a positive real. For $\mu,\nu\in\Sigma_{d,R}$ we have $W_{2}(\mu,\nu)\le B$: with $s=W_{2}(\mu,\nu)\ge0$, either $s\le1\le B$, or $1<s$ and then, multiplying by $s>0$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), $s<s^{2}\le4dR^{2}<B$ by The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points §bounded. Fix $k_{0}\in K$. For $\lambda,\lambda'\in\Sigma_{d,R}$, claims 1, 2 and 4 of The Distance to a Set is Nonexpansive give

and claim 5 of that lemma says that $D$ is uniformly continuous on $\Sigma_{d,R}$. In particular $D$ is bounded above and below.

Step 2 (Uniform continuity gives semicontinuity). Let $g:\Sigma_{d,R}\to\mathbb{R}$ be uniformly continuous on $\Sigma_{d,R}$ in the sense of Uniformly Continuous Map Between Metric Spaces. Given $\lambda\in\Sigma_{d,R}$ and $\varepsilon>0$, take the $\delta>0$ of that definition; if $W_{2}(\lambda,\lambda')<\delta$ then $|g(\lambda)-g(\lambda')|<\varepsilon$, hence $-\varepsilon<g(\lambda)-g(\lambda')<\varepsilon$, that is, $g(\lambda')<g(\lambda)+\varepsilon$ and $g(\lambda)-\varepsilon<g(\lambda')$ (claim 1 of Elementary Order Arithmetic in an Ordered Field). So $g$ is upper semicontinuous and lower semicontinuous on $\Sigma_{d,R}$ in the sense of Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space. By Step 1 this applies to $D$, and it applies to every $U$ as in claim 1.

Step 3 (Claim 1: existence). The hypotheses of Distance Functions Give Explicit Slope-Based Solutions of the Eikonal and Discounted Hopf-Lax Equations §eikonal hold for $X=\Sigma_{d,R}$, the metric $W_{2}$ and the nonempty set $K$ with $\Omega$ open, so the restriction of $D$ to $\Omega$ is an s-solution of $|\nabla u|=\mathbf{1}$ in $\Omega$.

Step 4 (Claim 1: the boundary condition). Let $U$ be as in claim 1. Let $\beta>0$; then $\beta/2>0$ and $\beta/2+\beta/2=\beta$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. First choose, by uniform continuity of $U$, a real $\sigma_{0}>0$ with $|U(\lambda)-U(\lambda')|<\beta/2$ whenever $W_{2}(\lambda,\lambda')<\sigma_{0}$; then let $\sigma$ be the lesser of $\sigma_{0}$ and $\beta/2$ (claim 9 of Elementary Order Arithmetic in an Ordered Field), a positive real with $\sigma\le\sigma_{0}$ and $\sigma\le\beta/2$. Let $\lambda,\lambda'\in\Sigma_{d,R}$ satisfy $D(\lambda)+D(\lambda')+W_{2}(\lambda,\lambda')<\sigma$. As $D(\lambda')$, $D(\lambda)$ and $W_{2}(\lambda,\lambda')$ are nonnegative, $D(\lambda)<\sigma$ and $D(\lambda')<\sigma$. Since $D(\lambda)$ is the infimum of the set $\{W_{2}(\lambda,k):k\in K\}$ by Distance from a Point to a Nonempty Subset of a Metric Space, a nonempty set bounded below by $0$, claim 2 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ gives $k\in K$ with $W_{2}(\lambda,k)<\sigma\le\sigma_{0}$; then $|U(\lambda)-U(k)|<\beta/2$, and $U(k)=0$, so $-\beta/2<U(\lambda)<\beta/2$. In the same way $-\beta/2<U(\lambda')<\beta/2$. Consequently

So the condition of Comparison Principle for Slope-Based Solutions of the Eikonal Equation on a Complete Metric Space with Interpolation Points §boundary holds both for the pair $(u,v)=(U,D)$ and for the pair $(u,v)=(D,U)$, with the same $\sigma$.

Step 5 (Claim 1: uniqueness). The constant function $\mathbf{1}$ is uniformly continuous on $\Omega$ (every difference of its values is $0$, so any $\delta>0$ works), and $c_{0}=1$ is a positive real with $c_{0}\le\mathbf{1}(\lambda)$ for $\lambda\in\Omega$. An s-solution is both an s-subsolution and an s-supersolution by Slope-Based Viscosity Subsolutions, Supersolutions and Solutions on a Metric Space §solution. Apply Comparison Principle for Slope-Based Solutions of the Eikonal Equation on a Complete Metric Space with Interpolation Points first with $u=U$ and $v=D$: both are bounded above and below (hypothesis and Step 1), $U$ is upper and $D$ lower semicontinuous on $\Sigma_{d,R}$ (Step 2), the restriction of $U$ to $\Omega$ is an s-subsolution (hypothesis) and that of $D$ an s-supersolution (Step 3), and the boundary condition holds by Step 4; its conclusion Comparison Principle for Slope-Based Solutions of the Eikonal Equation on a Complete Metric Space with Interpolation Points §comparison gives $U(\lambda)\le D(\lambda)$ for every $\lambda$. Apply it next with $u=D$ and $v=U$, whose hypotheses hold by the same steps; it gives $D(\lambda)\le U(\lambda)$. Hence $U=D$.

Step 6 (Claim 2). Let $\rho$, $b$ and $a$ be as in claim 2, and let $f:\Sigma_{d,R}\to\mathbb{R}$, $f(\lambda)=b\,D(\lambda)^{2}$. By Distance Functions Give Explicit Slope-Based Solutions of the Eikonal and Discounted Hopf-Lax Equations §discounted, applied to $X=\Sigma_{d,R}$ and the nonempty set $K$, we have $a\ge0$ and $a\,D^{2}$ is an s-solution of $\rho\,u+\frac{1}{2}|\nabla u|^{2}=f$ in $\Sigma_{d,R}$. It is bounded above and below: by Step 1 and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $0\le D(\lambda)^{2}\le B^{2}$, so $0\le a\,D(\lambda)^{2}\le aB^{2}$. Next, $f$ is uniformly continuous on $\Sigma_{d,R}$: for $\lambda,\lambda'$, the field identity $D(\lambda)^{2}-D(\lambda')^{2}=(D(\lambda)-D(\lambda'))(D(\lambda)+D(\lambda'))$, claims 1 and 4 of Properties of the Absolute Value in an Ordered Field and Step 1 give

so for $\varepsilon>0$ the positive real $\delta=\varepsilon/(2bB+1)$ satisfies $|f(\lambda)-f(\lambda')|<(2bB+1)\delta=\varepsilon$ whenever $W_{2}(\lambda,\lambda')<\delta$. Now let $U:\Sigma_{d,R}\to\mathbb{R}$ be bounded above and below and an s-solution of the same equation; by Slope-Based Viscosity Subsolutions, Supersolutions and Solutions on a Metric Space §solution both $U$ and $a\,D^{2}$ are s-subsolutions and s-supersolutions. Comparison Principle for Slope-Based Solutions of the Discounted Hopf-Lax Equation on a Complete Metric Space with Interpolation Points §comparison, whose hypotheses (completeness, interpolation points, uniform continuity of $f$, boundedness) hold, applied with $(u,v)=(U,a\,D^{2})$ gives $U\le a\,D^{2}$, and applied with $(u,v)=(a\,D^{2},U)$ gives $a\,D^{2}\le U$. Hence $U=a\,D^{2}$, and $a\,D^{2}$ is the only such function.

Step 7 (Claim 3: nonemptiness). Let $\Gamma$ be the set of commuting laws, and let $i,j\in[d]$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $c_{ij}=x_{(i)(j)}-x_{(j)(i)}$, where $(i)(j)$ and $(j)(i)$ are words of length $2$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, hence different from $\varnothing$, which has no length by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §words. So the coefficients of both monomials at $\varnothing$ are $0$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials, and $c_{ij}(\varnothing)=0$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear. Since the only factorisation of $\varnothing$ is $(\varnothing,\varnothing)$ by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §factorisations, the definition of the product in The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §product (a sum over a one-element set, equal to its single term by Sum over a Finite Index Set and claim 1 of Properties of Finite Sums) gives

The map $\delta(p)=p(\varnothing)$ belongs to $\Sigma_{d,R}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §zero-law, and $\delta(c_{ij}^{*}c_{ij})=0$ for all $i,j$; so $\delta\in\Gamma$ and $\Gamma$ is nonempty.

Step 8 (Claim 3: weak-star limits). Let $(\lambda_{m})$ be a sequence in $\Gamma$ and $\lambda\in\Sigma_{d,R}$ with $\lambda_{m}\to\lambda$ weak-star, and fix $i,j\in[d]$ and $q=c_{ij}^{*}c_{ij}$. Since $\lambda_{m}(q)=0$, the real sequences $(\operatorname{Re}\lambda_{m}(q))_{m}$ and $(\operatorname{Im}\lambda_{m}(q))_{m}$ are constantly $0$ by the uniqueness in Real and Imaginary Parts of a Complex Number, so they converge to $0$; by Weak-Star Convergence of Noncommutative Laws §weak-star they also converge to $\operatorname{Re}\lambda(q)$ and $\operatorname{Im}\lambda(q)$. Claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences gives $\operatorname{Re}\lambda(q)=\operatorname{Im}\lambda(q)=0$, so $\lambda(q)=0$ by Real and Imaginary Parts of a Complex Number. As $i,j$ were arbitrary, $\lambda\in\Gamma$. Thus $\Gamma\subseteq\Sigma_{d,R}$ is nonempty and contains every $\lambda\in\Sigma_{d,R}$ that is a weak-star limit of a sequence in $\Gamma$, which are exactly the hypotheses imposed on $K$; so claims 1 and 2 apply with $K=\Gamma$. $\blacksquare$