Comparison and uniqueness for viscosity solutions of first-order Hamilton-Jacobi equations in Banach spaces without smooth norms: the Crandall-Lions problem for $L^1$, $L^\infty$, spaces of continuous functions, and spaces of measures

The viscosity theory of first-order Hamilton-Jacobi equations $u+H(x,Du)=0$ in infinite-dimensional Banach spaces was initiated by Crandall and Lions in 1985: they proved comparison and uniqueness in spaces with the Radon-Nikodym property via Stegall's perturbed-optimization lemma, introduced strict viscosity solutions in an appendix, and in Parts II-III (1986) added existence, moduli of continuity, and a comparison theory for spaces with differentiable norms via Ekeland's variational principle. The same series states its own boundary: the smooth-norm requirement 'excludes extremely important choices for $V$ like $L^1$, $L^\infty$, spaces of continuous functions and the space of bounded measures', where 'the study of equations in these spaces remains almost totally open', and an explicit $\ell^1$ counterexample (a space with the Radon-Nikodym property but no smooth norm) shows comparison fails for the standard and strict notions there. Forty years later, whether a comparison/uniqueness theory under norm geometry exists for the excluded spaces, and which Banach spaces admit such a theory, remains open: the modern lines — beta-viscosity solutions (requiring a beta-smooth norm), metric viscosity solutions, and Hamilton-Jacobi-Bellman comparison on Wasserstein spaces — either restrict to smooth norms or change the geometry. The excluded spaces are the natural state spaces for stochastic control under partial observation and measure-valued filtering, so the gap is substantive rather than merely technical.

Background

First-order Hamilton-Jacobi equations of the form $u+H(x,Du)=0$, where $H$ is a continuous Hamiltonian and $Du$ denotes the gradient of the unknown function $u$, are the dynamic-programming (Bellman) equations of deterministic optimal control: the value function of a control problem formally satisfies such an equation. Classical solutions generally do not exist because solutions develop singularities, and the successful framework is the theory of viscosity solutions: $u$ is a viscosity subsolution if whenever a smooth test function touches $u$ from above at $x$, the equation holds with the test function's gradient in place of $Du(x)$ (supersolutions are defined symmetrically). All of the finite-dimensional theory (comparison and uniqueness, stability, existence via Perron's method) rests on local compactness, which disappears when the state variable $x$ lives in an infinite-dimensional Banach space $V$ — precisely the setting of optimal control of evolution equations (for example control of PDEs), where the value function is defined on an infinite-dimensional state space.

Crandall and Lions began the infinite-dimensional theory in the series 'Hamilton-Jacobi equations in infinite dimensions' (Parts I-VII, 1985-1994). Part I (J. Funct. Anal. 62 (1985) 379-396) proves comparison and uniqueness for bounded uniformly continuous $u$ when $V$ has the Radon-Nikodym property (RNP: every Lipschitz curve in $V$ is differentiable almost everywhere; reflexive spaces and separable duals have it, while $L^1$, $L^\infty$, spaces of continuous functions, and spaces of measures do not). Because maxima over non-compact sets need not be attained, they replace compactness by Stegall's lemma (in an RNP space, closed bounded sets admit continuous linear functionals of arbitrarily small norm that attain their supremum on the set), perturbing the functional to be maximized. The doubling-of-variables argument (the standard technique for comparing a subsolution and a supersolution by maximizing their difference over pairs of points, using the same auxiliary function in both variables) additionally requires auxiliary functions $d$ and $v$ with controlled differentiability — the 'conditions (C)' — and for which spaces such functions exist is itself a geometric question. The appendix of Part I introduces strict viscosity solutions, defined through the approximate Frechet sub- and superdifferentials $D_\varepsilon^\pm u(x)$ (the set of slopes admitting a first-order expansion of $u$ at $x$ up to error $\varepsilon$ in a neighborhood), a notion designed to bypass smooth test functions.

Part II (J. Funct. Anal. 65 (1986) 368-405) supplies existence of viscosity solutions for the Cauchy and stationary problems under conditions (C) via differential games, and its Section 2 proves the promised moduli of continuity in the (C) setting, which includes Hilbert spaces. Part III (J. Funct. Anal. 68 (1986) 214-247) develops strict viscosity solutions into a first-class notion: they are stable under local uniform convergence and coincide with the standard notion whenever the norm of $V$ is Frechet-differentiable away from the origin, and, using Ekeland's variational principle (a perturbed-optimization statement valid in every Banach space, unlike Stegall's lemma) together with a norm-like function $N$ that is Lipschitz, differentiable off the origin, and dominates the norm, it proves comparison, uniqueness and existence for the model stationary and Cauchy problems in such spaces — a class that includes some spaces without the RNP but provably excludes $L^1$, $L^\infty$, spaces of continuous functions, and spaces of measures, since none of these admits an equivalent smooth norm. Part III also shows that the obstruction is geometric rather than measure-theoretic: an explicit counterexample in $\ell^1$ (which does have the RNP but no smooth norm) shows comparison fails for strict viscosity solutions there. Parts IV-V (1990, 1991) extend the theory to Hamiltonians with unbounded linear terms, using a B-continuity condition (continuity adapted to the unbounded setting).

The interest of the excluded spaces is not incidental: spaces of probability measures are the natural state spaces for control under partial observation (measure-valued filtering), the authors' stated motivation. The later frameworks change the problem's geometry rather than solving the original question: beta-viscosity solutions of Tran Van Bang and Phan Trong Tien replace Frechet differentiability by a weaker beta-derivative (a derivative notion in which the usual balls controlling the remainder are replaced by the sets of a chosen bornology, a family of bounded sets, on the Banach space) and prove existence, uniqueness and stability, but still require a beta-smooth equivalent norm and prove failure of comparison without it; metric viscosity solutions of Gangbo and Swiech, whose definition (based on the local slope $|\nabla u|(x)=\limsup_{y\to x}(u(x)-u(y))^+/d(x,y)$) they state was influenced by the strict viscosity solutions of Part III, are executed for length/geodesic — in practice Hilbert- and Wasserstein-type — metric spaces; and the recent comparison principles on spaces of probability measures (Conforti-Kraaij-Tonon for controlled gradient flows; Daudin-Seeger, exploiting differentiability of the 2-Wasserstein distance (the optimal-transport distance between probability measures) with an entropy penalization; Bertucci-Lions, extending Stegall's lemma to Wasserstein spaces at the price of a nonlinear perturbation) all exploit the Wasserstein geometry of measures, not the Banach-norm geometry of the underlying space of measures.

Problem Statement

The surviving open core of the Crandall-Lions program (Parts I-III). For first-order Hamilton-Jacobi equations $u+H(x,Du)=0$ and the corresponding Cauchy problem in a Banach state space $V$ that admits no equivalent smooth norm — in particular $L^1$, $L^\infty$, spaces of continuous functions, and spaces of bounded measures, which are the mandatory target — establish a comparison and uniqueness (well-posedness) theory for viscosity solutions in the original framework based on the norm topology, or prove that none is possible for any admissible solution notion based on that topology. A norm-topology solution notion is admissible when it agrees with the classical viscosity notion whenever the norm of $V$ is smooth, is stable under local uniform convergence, and has a non-degenerate solution class, one containing the bounded uniformly continuous solutions of the model Cauchy and stationary problems. The question is settled once each of the four named spaces is resolved: for each one, either prove that every admissible norm-topology solution notion fails comparison and uniqueness, or exhibit an admissible norm-topology solution notion for which comparison and uniqueness hold; the answer may differ from one named space to another. The known boundary is one-sided: the Radon-Nikodym property alone is insufficient (comparison fails in $\ell^1$ by the Part III counterexample), while the positive theories cover Hilbert spaces, RNP spaces with differentiable norm, and beta-smooth-norm spaces.

References

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2. Michael G. Crandall and Pierre-Louis Lions, 'Hamilton-Jacobi Equations in Infinite Dimensions. Part 1. Uniqueness of Viscosity Solutions', Mathematics Research Center Technical Summary Report #2769, November 1984, DTIC accession ADA149094, https://archive.org/details/DTIC_ADA149094
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