Shifted-center radial test functions in the definition of B-continuous viscosity solutions for Hamilton-Jacobi equations with unbounded linear terms in Hilbert spaces

The Crandall-Lions definition of a B-continuous viscosity solution of a first-order Hamilton-Jacobi equation with an unbounded linear term in a separable Hilbert space, $u+\langle Ax,\nabla u\rangle+F(x,\nabla u)=0$ , tests the subsolution inequality against regular test functions and radial functions $g(x)=g_0(|x|)$ centered at the origin (the 0-formulation). Whether the definition is equivalent to the stronger, translation-covariant formulation in which the radial center may be shifted to an arbitrary point $x_0\in D(A)$ is known only when the unbounded linear operator $A$ is self-adjoint (and, in the weakly continuous framework, for centers in $D(A^*)$ ); the case of general non-self-adjoint maximal monotone $A$ has been open since the 1990-1991 installments of the Crandall-Lions series. The B-continuous framework became the standard treatment of Hamilton-Jacobi-Bellman equations in infinite dimensions (differential games, Perron's method, integro-PDE, path-dependent and mean-field equations), and all of these successors still use the 0-centered formulation; the most recent work (2025-2026) identifies the shifted-center obstruction as the essential remaining difficulty but circumvents it only in a different, second-order, B-continuity-free framework. No resolution of the first-order shifted-center question is known in the literature to date.

Background

Finite-dimensional background. For a first-order Hamilton-Jacobi equation $u+H(x,\nabla u)=0$ on $\mathbb{R}^n$ , a viscosity subsolution is an upper semicontinuous function $u$ such that $u(x_0)+H(x_0,D\varphi(x_0))\le 0$ whenever $\varphi$ is $C^1$ and $u-\varphi$ has a local maximum at $x_0$ ; the definition, introduced by Crandall and Lions in the 1980s, uses only first-order test functions and is stable under local uniform limits, which made it the standard weak-solution notion for Hamilton-Jacobi and Hamilton-Jacobi-Bellman equations (see the User's Guide of Crandall, Ishii and Lions). Infinite-dimensional setting. For optimal control of evolution equations in a real separable Hilbert space $H$ (for example, delay equations rewritten as evolution equations, or stochastic PDEs), the state equation contains an unbounded linear operator $A$ (for example, the negative Laplacian), so the Hamiltonian contains the unbounded linear term $\langle Ax,p\rangle$ . The naive infinite-dimensional extension of the viscosity definition fails: a Hilbert space admits no $C^1$ functions with bounded support, hence not enough test functions, and the pairing $\langle Ax,D\varphi(x)\rangle$ is undefined unless $D\varphi(x)\in D(A^*)$ , the domain of the adjoint $A^*$ . In their series of papers (Parts I-III, 1985-1986, under the title Hamilton-Jacobi equations in infinite dimensions; Parts IV-VII, 1990-1994, under the title Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions), Crandall and Lions resolved this by restricting the test functions to a class of regular functions $\varphi$ , weakly sequentially lower semicontinuous and such that $\nabla\varphi$ and $A^*\nabla\varphi$ are continuous, and by adjoining radial functions $g(x)=g_0(|x|)$ with $g_0$ nondecreasing and $C^1$ ; the monotonicity of $A$ (that is, $\langle Ax-Ay,x-y\rangle\ge 0$ , which for linear $A$ gives $\langle Ax,x\rangle\ge\langle A0,x\rangle=0$ ) makes the radial contribution absorb the unbounded term. In this definition all radial test functions are centered at the origin: the 0-formulation. B-continuity (Part V, 1991). Norm continuity is too strong and weak continuity too weak for the comparison principle in this setting, so Part V introduced B-continuity: $u$ is B-continuous if $u(x_n)\to u(x)$ whenever $x_n\rightharpoonup x$ weakly and $Bx_n\to Bx$ in norm, where $B$ is a bounded, strictly positive, self-adjoint operator with $A^*B$ bounded. The strong B-condition $\langle(A^*B+C_0B)x,x\rangle\ge|x|^2$ (respectively the weak B-condition with right-hand side $0$ ) ties $B$ to $A$ ; a result of Renardy (1995), which he formulated as solving a problem of Crandall and Lions, shows that such a $B$ always exists in the weak case, for instance $B=((A+I)(A^*+I))^{-1/2}$ . The B-continuous framework of Part V, with its comparison principle, existence theory (via a differential-game representation: the solution is the value of a zero-sum game and satisfies the corresponding Isaacs equation, the Hamilton-Jacobi equation of that game) and stability results, is the foundation of the modern theory of second-order Hamilton-Jacobi-Bellman equations in Hilbert spaces presented in the monograph of Fabbri, Gozzi and Święch, which is the standard tool for stochastic optimal control of delay equations and SPDEs. The question. The 0-formulation is tied to the origin, and it is not known whether the radial test functions may instead be centered at an arbitrary point $x_0\in D(A)$ . For centers in $D(A^*)$ the corresponding translation-covariant formulation is equivalent to the 0-formulation (Part IV proved this for weakly semicontinuous functions, $D(A^*)$ being the natural domain since the pairing $\langle Ax,\nabla\varphi(x)\rangle=\langle x,A^*\nabla\varphi(x)\rangle$ involves $A^*$ ), and for centers in $D(A)$ the equivalence is proved when $A$ is self-adjoint (then $D(A)=D(A^*)$ ), by a Yosida-approximation argument through the resolvent $J_\varepsilon=(I+\varepsilon A)^{-1}$ , which maps $H$ into $D(A)$ and tends to the identity as $\varepsilon\to 0$ ; the non-self-adjoint case is the open problem. Its importance is threefold: (a) it decides whether the B-continuous solution concept is a genuine translation-covariant weak-solution notion rather than an artifact of the origin; (b) a larger class of admissible test functions directly strengthens uniqueness, comparison and stability arguments, which consume test functions; (c) the same obstruction, namely a fixed shift interacting badly with the unbounded linear term, is what the newest second-order theory still has to circumvent: Tang and Zhou (2026) observe that in the monograph framework only the unshifted norm can be added to the test functions because the shifted norm $|\cdot-y|_H$ with $y\neq 0$ fails the required Itô inequality (the sufficient condition under which Itô's formula applies to a candidate test function inside the doubling-of-variables comparison argument), and call this fact the essential difficulty of the theory of viscosity solutions to HJB equations with unbounded linear term.

Problem Statement

Let $H$ be a real separable Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$ , let $A\colon D(A)\subset H\to H$ be a densely defined linear maximal monotone operator, that is, $\langle Ax-Ay,x-y\rangle\ge 0$ for all $x,y\in D(A)$ , admitting no proper monotone linear extension (successor presentations in this line, including the Fabbri-Gozzi-Święch monograph, state the standing hypotheses on $-A$ and call $A$ maximal dissipative; that is the same framework with the sign reversed, not an additional assumption), and let $A^*$ denote the adjoint of $A$ . Let $B$ be a bounded, strictly positive, self-adjoint operator on $H$ with $A^*B$ bounded, satisfying the strong B-condition $\langle(A^*B+C_0B)x,x\rangle\ge|x|^2$ for some constant $C_0$ (or, alternatively, the weak B-condition with right-hand side $0$ ). Let $F$ be a continuous Hamiltonian, $F\colon H\times H\to\mathbb{R}$ for the stationary equation and $F\colon[0,T]\times H\times H\to\mathbb{R}$ for the evolutionary one, satisfying the standing structure conditions that Parts IV-V of the Crandall-Lions series impose on the continuity and growth of $F$ . Consider the first-order Hamilton-Jacobi equation $u(x)+\langle Ax,\nabla u(x)\rangle+F(x,\nabla u(x))=0$ on $H$ , together with its evolutionary version $u_t+\langle Ax,\nabla u\rangle+F(t,x,\nabla u)=0$ on $[0,T]\times H$ . A function $u$ is B-continuous if $u(x_n)\to u(x)$ whenever $x_n\rightharpoonup x$ weakly in $H$ and $Bx_n\to Bx$ in norm. In the 0-formulation, a B-continuous viscosity subsolution $u$ is required to satisfy the following: at every local maximum point $z$ of $u-\varphi-g$ , where $\varphi$ ranges over the regular test functions of the source framework (weakly sequentially lower semicontinuous $\varphi$ with $\nabla\varphi$ and $A^*\nabla\varphi$ continuous) and $g(x)=g_0(|x|)$ ranges over the nondecreasing $C^1$ radial functions centered at the origin, the subsolution inequality holds, with the unbounded linear term evaluated through the adjoint on the regular part alone and the radial gradient entering only through the Hamiltonian, namely $u(z)+\langle z,A^*\nabla\varphi(z)\rangle+F(z,\nabla\varphi(z)+\nabla g(z))\le 0$ , with the time-dependent analogue at local maxima of $u-\varphi-g$ on $[0,T]\times H$ . Question: does the 0-formulation imply the stronger, translation-covariant formulation in which the radial test functions may be centered at an arbitrary point of $D(A)$ ; that is, must every B-continuous viscosity subsolution in the 0-formulation also satisfy the above subsolution inequality at every local maximum of $u-\varphi-g_0(|\cdot-x_0|)$ , for every $x_0\in D(A)$ (the converse direction being trivial since $0\in D(A)$ )? Equivalently: do the two formulations define the same class of B-continuous subsolutions, supersolutions (defined by the sign-reversed inequalities, with nonincreasing radial functions, at local minima), and solutions of the stationary and evolutionary equations? The implication is known when $A$ is self-adjoint (then $D(A)=D(A^*)$ , and the Yosida-approximation argument of Part IV applies) and, in the weakly continuous framework of Part IV, for centers in $D(A^*)$ ; the case of a general non-self-adjoint maximal monotone $A$ with centers in $D(A)$ is what is open.

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