The unresolved core of the Crandall-Lions existence theory for viscosity solutions of Hamilton-Jacobi equations in infinite dimensions: convergent finite-dimensional approximation, non-smooth Banach spaces, boundary value problems, and growth at infinity

Crandall and Lions (J. Funct. Anal. 65 (1986) 368-405) proved existence - completing the uniqueness theory of Part I - of viscosity solutions for first-order Hamilton-Jacobi equations in Banach spaces with the Radon-Nikodym property, for uniformly continuous data and Hamiltonians satisfying structural conditions, and stated the directions their theory did not cover. The same authors' Parts III-VI (1986-1993) and Tataru's related work settled two of these directions: Hamiltonians with unbounded linear or nonlinear operator terms, and removal of the Radon-Nikodym property via strict viscosity solutions. Four decades later the remaining core is still open: (i) a convergence theory for finite-dimensional (Galerkin-type) approximations, whose naive version provably fails and for which no general convergent alternative is known; (ii) well-posedness in Banach spaces without a differentiable norm-like function, such as $L^1$, $L^\infty$, $C(K)$, and spaces of measures; (iii) a general theory of boundary value problems; and (iv) solution classes with behavior at infinity beyond uniform continuity; together with sharp delimitation of the structural hypotheses. This problem asks for existence, uniqueness/comparison, and convergent-approximation results covering these excluded regimes.

Background

Hamilton-Jacobi equations are first-order nonlinear partial differential equations of the general evolution form $u_t + H(x,t,u,Du) = 0$ or stationary form $u + H(x,u,Du) = 0$, where $u$ is an unknown real-valued function and $Du$ denotes its gradient (in the settings below, the Fréchet derivative on a Banach space). They arise principally as the Hamilton-Jacobi-Bellman (HJB) equations of dynamic programming in optimal control: $u$ is the value function of a control problem and $H$ encodes the controlled dynamics and running cost. Classical solutions typically cease to exist once characteristics cross, and the modern theory treats these equations in the framework of viscosity solutions: weak solutions defined by requiring that whenever $u - \phi$ has a local maximum (respectively minimum) at a point, for a smooth test function $\phi$, the appropriate one-sided inequality for $H$ holds at that point. This notion requires no differentiability of $u$ and is stable under uniform limits, which makes existence and uniqueness questions meaningful under quite general hypotheses.

When the state of the controlled system is itself distributed - the optimal control of evolution partial differential equations such as heat or wave equations, and, more recently, mean-field control problems on spaces of probability measures - the HJB equation lives on an infinite-dimensional Banach or Hilbert space $V$. Infinite dimensions remove the basic tools of the finite-dimensional theory: bounded sets are not precompact, so Arzelà-Ascoli-type compactness fails; there is no translation-invariant reference measure and no canonical Laplacian, so vanishing-viscosity and distributional regularization are unavailable; and in many function spaces the norm is nowhere Fréchet differentiable, so there may be no smooth gauge of distance. In addition, the dynamics of distributed systems is generated by unbounded linear operators $A$ (defined on dense domains, discontinuous in every reasonable topology), and the Hamiltonian then contains the singular term $(Ax, Du)$ pairing the unbounded vector $Ax$ with the gradient.

Crandall and Lions developed the infinite-dimensional theory in a series of seven papers between 1985 and 1994. Part I (J. Funct. Anal. 62 (1985) 379-396) proved uniqueness by adapting the "doubling of variables" technique - testing the equation satisfied by one solution against the other at pairs of points, coupled by a penalization of the distance between them - to infinite dimensions. The crucial analytic device is a variational principle that makes a perturbed supremum be attained; Part I used Stegall's principle, which requires $V$ to have the Radon-Nikodym property (RNP), a geometric property of Banach spaces - equivalent, for this purpose, to every Lipschitz curve in $V$ being differentiable almost everywhere - which holds in reflexive spaces and fails in many classical non-reflexive ones. Part II, the source of this problem, proved existence in the same setting: a real Banach space with the RNP, conditions (C) on auxiliary functions $d : V \times V \to [0,\infty)$ and $v : V \to [0,\infty)$ controlling the metric and the growth of the Hamiltonian, and structural hypotheses (H0)-(H4) on $H$; the unique viscosity solution is uniformly continuous (on $V \times [0,T]$ for the evolution problem) and satisfies a growth bound $|u| \le A + Bv$. The proof passes to limits of approximate problems, establishes a priori modulus estimates, reduces general equations to model cases, and, for Lipschitz $H$, represents the solution as the lower value of a differential game. The paper stated that the theory left open at least: Hamiltonians with unbounded linear terms; the question of finite-dimensional approximation by the Faedo-Galerkin scheme, whose general failure the series then demonstrated; Banach spaces without the RNP, where an $\varepsilon$-formulation of the equation had to be used; the lack of a characterization of when existence and uniqueness hold, with counterexamples showing that hypotheses such as (H2) and (H4) could not simply be dropped; boundary value problems; and solution classes with behavior at infinity other than uniform continuity.

The series itself then resolved part of this list. Part III (J. Funct. Anal. 68 (1986) 214-247) introduced strict viscosity solutions - where the test-function inequalities must hold up to a fixed margin - using Ekeland's variational principle (which yields only approximate minimizers on general complete metric spaces and hence forces stricter inequalities) instead of Stegall's; this removed the RNP at the price of requiring a differentiable norm-like function on the space. Part III also proved existence for coercive Hamiltonians under a weakened condition, treated certain equations with unbounded operators arising in the control of evolution equations, and exhibited the examples showing the "dramatic failure" of the basic Faedo-Galerkin method. (The Faedo-Galerkin scheme projects the equation onto an increasing sequence of finite-dimensional subspaces $V_d$ of $V$; the structural obstruction is that the projection $P_d$ onto a proper subspace satisfies $\|I - P_d\| = 1$ however large $d$ is, so the projected equations are never uniformly consistent with the original one.) Part IV (1990) handled Hamiltonians with unbounded linear terms for solutions that are weakly sequentially continuous; Part V (1991) treated B-continuous solutions - continuous on bounded sets for the bounded-weak topology, the strongest topology that agrees with the weak topology on bounded sets; Part VI (1993) extended the theory to nonlinear accretive operators $A$ (an operator is $m$-accretive when it generates a nonexpansive semigroup on $V$), refining Tataru's method and adding Perron-type existence, which constructs the solution as the supremum of the family of subsolutions. Independently, Tataru (1992, 1994) proved existence and uniqueness for Hamiltonians with unbounded nonlinear terms. Part VII (1994) added a fundamental delimitation: the value function of an infinite-dimensional optimal control problem does not in general satisfy the associated HJB equation even in the viscosity sense, so dynamic programming itself has limits in infinite dimensions.

Part III's strict-viscosity framework still needs a differentiable norm-like function on the ambient space; a counterexample in $\ell^1$ (recorded in the series) shows the framework fails without one, leaving the classical spaces $L^1$, $L^\infty$, $C(K)$, and spaces of measures without a complete theory. Later frameworks cover parts of this ground: the beta-viscosity solutions of Tran and Phan (2020), based on a derivative notion for functions of low regularity on Banach spaces, and the metric viscosity solutions of Ambrosio and Feng (2014) and Gangbo and Święch (2015) on length spaces and on the Wasserstein space of probability measures. The monograph of Fabbri, Gozzi, and Święch (2017) organizes the by-then-standard B-continuity framework and lists finite-dimensional approximation as one of the three standard existence techniques - alongside the dynamic programming principle and Perron's method - while noting that each has limitations of its own. On the computational side, discretizing equations on infinite-dimensional spaces remains an acknowledged bottleneck: the machine-learning schemes of Cohen, de Feo, Hebner, and Sirignano (2026) for PDEs and optimal control on Hilbert spaces avoid projection altogether precisely because, as their authors state, proving convergence of the discretized problems to the limit problem "is not well-understood". In the parallel setting of HJB equations on the Wasserstein space, Gangbo, Mayorga, and Święch (2021) and Mayorga and Święch (2023) proved convergence of finite-dimensional particle approximations, with error estimates by Daudin, Jackson, and Seeger (2025) - an active frontier that shows both that the approximation question is the live one and that no general Hilbert/Banach-space theory exists. What remains, four decades after Part II, is the subject of this problem: a well-posedness-and-approximation theory for the regimes the Crandall-Lions series excluded - convergent discretization, non-smooth Banach spaces, boundary value problems, and growth beyond uniform continuity.

Problem Statement

Let $V$ be an infinite-dimensional Banach space and consider the first-order Hamilton-Jacobi equations

$$u_t + H(x,t,u,Du) = 0 \quad \text{in } V \times (0,T], \qquad u(\cdot,0) = \phi$$

(evolution form) and

$$u + H(x,u,Du) = 0 \quad \text{in } V$$

(stationary form), where $Du$ is the Fréchet derivative, $H : V \times [0,T] \times \mathbb{R} \times V^* \to \mathbb{R}$ is a continuous Hamiltonian (with the $t$-variable omitted in the stationary form), and $\phi : V \to \mathbb{R}$ is initial data. The Crandall-Lions theory of 1985-1986 establishes existence and uniqueness of uniformly continuous viscosity solutions when $V$ has the Radon-Nikodym property and $H$ satisfies conditions (C) and (H0)-(H4); the continuation of the series through 1994, together with Tataru's work, then removed the unbounded-operator restriction (in adapted continuity classes) and, via strict viscosity solutions, the Radon-Nikodym property itself (in spaces admitting a differentiable norm-like function). The problem is to complete the well-posedness and approximation theory in the regimes still excluded, namely:

(1) Convergent finite-dimensional approximation. The naive Faedo-Galerkin scheme is known to fail in general. Establish conditions on the equation, the data, and the approximation scheme under which finite-dimensional approximations of the equation converge to the unique viscosity solution of the infinite-dimensional equation - or construct provably convergent alternative discretizations, with convergence rates where possible - and delimit the known counterexamples; in particular, characterize what can overcome the obstruction $\|I - P_d\| = 1$ for projections $P_d$ onto finite-dimensional subspaces.

(2) Non-smooth Banach spaces. Establish existence, uniqueness, and comparison for viscosity solutions in Banach spaces that admit no differentiable norm-like function - for example $L^1$, $L^\infty$, $C(K)$, and spaces of measures - where the strict-viscosity framework is known to fail (the $\ell^1$ counterexample).

(3) Boundary value problems. Develop a general well-posedness theory (existence, uniqueness/comparison) for boundary value problems for these equations on domains of $V$ with boundary data - going beyond the special classes treated so far (exit-time problems, state-constrained control, boundary control) - and covering Hamiltonians with unbounded terms.

(4) Growth at infinity. Prove existence and uniqueness for solution classes whose behavior at infinity goes beyond the uniformly continuous, $v$-growth class of the Crandall-Lions theory (unbounded value functions, polynomial or faster growth), identifying the modified structural conditions under which comparison remains valid.

(5) Sharpness of hypotheses. For each regime above, determine whether the structural conditions imposed - in the spirit of the (H2)/(H4) necessity counterexamples of the original theory - are necessary, so that the obtained existence and uniqueness conditions are sharp or near-sharp.

The common objective is a well-posedness-and-approximation theory for first-order infinite-dimensional Hamilton-Jacobi equations covering the settings excluded by the Crandall-Lions series; components (1)-(4) each address one excluded regime and component (5) delimits all of them. Each of components (1)-(4) has independent standing - it is separately refutable and can be resolved on its own: resolving a single regime, together with its sharpness question in (5), settles that regime while the other regimes and their sharpness questions remain open. The question concerns the four regimes (1)-(4), each together with its sharpness question in (5).

References

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