Equivalence of the Crandall-Lions 0-centered and D(A)-centered formulations of viscosity solutions for non-self-adjoint maximal monotone linear operators in Hilbert spaces

For Hamilton-Jacobi equations in infinite-dimensional separable Hilbert spaces containing the unbounded linear term $⟨Ax, Du⟩$ (with $A$ a linear, densely defined, maximal monotone operator, such as a spatial differential operator), Crandall and Lions (Part IV, 1990) introduced a viscosity solution concept in which test functions split into a 'good part' $φ$, whose gradient lies in the domain $D(A^*)$ of the adjoint so that the pairing $⟨Ax, ∇φ(x)⟩$ is meaningful, and a radial 'bad part' $g$, nondecreasing in the distance from the origin, whose pairing with $A$ is discarded using monotonicity. Part IV proves that, for weakly sequentially upper semicontinuous functions, this 0-formulation is equivalent to the stronger translation-invariant formulation in which the radial part is centered at an arbitrary point of $D(A^*)$; for centers in $D(A)$ the equivalence was proved only when $A$ is self-adjoint, and the general case was left open. The open core: is the 0-formulation equivalent to the formulation with radial parts centered at $x_0 ∈ D(A)$ for general (non-self-adjoint) linear maximal monotone $A$? The obstacle is that the center term $⟨Ax, x_0⟩$ is well defined but generally discontinuous in $x$. No resolution is known in the literature: the modern standard definition (B-continuous solutions in the Fabbri-Gozzi-Święch framework) retains radial parts centered at 0; the adjacent operator-theoretic problem posed by Crandall and Lions was solved by Renardy (1995) but is a different question; and the authors' own Part VI (1994) settled the nonlinear-operator extension.

Background

A Hamilton-Jacobi (HJ) equation is a first-order nonlinear partial differential equation of the form $u_t + H(x, t, u, Du) = 0$ (or, in stationary form, $u + H(x, u, Du) = 0$), where $Du$ is the spatial gradient and $H$ is the Hamiltonian; when the equation comes from an optimal control problem via dynamic programming it is called a Hamilton-Jacobi-Bellman (HJB) equation. Classical solutions typically fail to exist because characteristics cross, and the modern replacement is the viscosity solution (introduced by Crandall and Lions, 1983): an (upper or lower semi)continuous function $u$ that satisfies the differential inequality in a weak 'touching' sense, meaning every smooth test function that touches $u$ from above (resp. below) at a point must itself satisfy the subsolution (resp. supersolution) inequality there. This notion made first-order PDE control theory rigorous in finite dimensions.

In the optimal control of partial differential equations (for example, a heat or transport equation with boundary control), the state lives in an infinite-dimensional separable Hilbert space $H$ — a complete inner-product space with a countable orthonormal basis, such as $L^2$ — and the state equation is driven by an unbounded linear operator $A$ defined only on a dense domain $D(A) ⊂ H$ (e.g., a spatial derivative). The associated HJB equation then contains the unbounded linear term $⟨Ax, Du⟩$, which is not a meaningful classical expression: $Du$ need not exist, and even for smooth functions the pairing is defined only if the gradient meets the domain of the adjoint $A^*$ (the adjoint of $A$, defined on its own domain $D(A^*)$).

Crandall and Lions resolved this in their series 'Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions' (Parts I-VII), with Part IV (1990) devoted to Hamiltonians with unbounded linear terms. Their test functions split as $ψ = φ + g$: a 'good part' $φ$, regular enough that $∇φ(x̄) ∈ D(A^*)$ and $A^*∇φ$ is continuous, so the pairing is defined as $⟨Ax̄, ∇φ(x̄)⟩ := ⟨x̄, A^*∇φ(x̄)⟩$; and a 'bad part' $g$ of class $C^1$ that is radial about the origin ($g(x) = g̃(‖x‖)$) and nondecreasing in the radius, whose contribution $⟨Ax, ∇g(x)⟩$ is discarded using monotonicity. Here a linear operator $A$ is monotone if $⟨Ax, x⟩ ≥ 0$ for all $x ∈ D(A)$, and maximal monotone if it admits no proper monotone linear extension; $A$ is self-adjoint if $A = A^*$. Since $g′(r) ≥ 0$ and $⟨Ax, x⟩ ≥ 0$, dropping the bad part's pairing preserves the direction of the inequality. The theory assumes a bounded, positive (that is, $⟨Bx, x⟩ ≥ 0$ for all $x$), self-adjoint operator $B$ on $H$ with $A^*B$ bounded, satisfying either the strong B-condition $⟨(A^*B + C_0B)x, x⟩ ≥ ‖x‖²$ or the weak B-condition $⟨(A^*B + C_0B)x, x⟩ ≥ 0$ (Part IV conditions (1.12) and (1.12)_w); Part V (1991) supplemented this with B-continuity (continuity with respect to the norm induced by $B$) of the solution class.

Within this framework Part IV introduced two a priori different notions. Definition 1.1 (the 0-formulation) takes the radial bad part centered at the origin. Definition 1.1' (the $x_0$-formulation) allows the radial bad part to be centered at an arbitrary point $x_0$ — a translation-invariant, more local notion, closer in spirit to finite-dimensional viscosity testing. The appendix of Part IV (Proposition A.1) proves that, for weakly sequentially upper semicontinuous functions (semicontinuous along weakly convergent sequences), the two formulations are equivalent when the allowed centers lie in $D(A^*)$, for any maximal monotone $A$; but for centers $x_0 ∈ D(A)$ the equivalence was established only when $A$ is self-adjoint. The obstruction is instructive: for $x_0 ∈ D(A)$ the center contributes the term $⟨Ax, x_0⟩$, which is well defined (both arguments lie in $H$) but generally discontinuous in $x$; when $x_0 ∈ D(A^*)$ it rewrites as $⟨x, A^*x_0⟩$ and when $A$ is self-adjoint as $⟨x, Ax_0⟩$ — both continuous — while in general no mechanism is known to control it.

Tataru (1992, 1992, 1994) introduced alternative definitions of viscosity solutions for equations with unbounded (including nonlinear) terms; Ishii (1993) developed the second-order Hilbert-space theory; Part VI (1994) of the series extended the theory to nonlinear unbounded operators by refining Tataru's method; Part VII (1994) showed that the value function does not always satisfy the HJB equation under these hypotheses; Renardy (1995) solved a different operator-theoretic problem posed by Crandall and Lions (constructing polar decompositions with range inclusion $R(B^{1/2}) ⊆ D(A^*)$); the books of Li and Yong (1995) and Fabbri, Gozzi and Święch (2017, Definition 3.35) standardized the modern presentation, which still centers the radial test part at the origin even for general maximal dissipative $A$ (maximal dissipative: $-A$ is maximal monotone, the same class of operators under the opposite sign convention); and recent papers (Wessels 2023; de Feo-Święch-Wessels 2023; Święch-Wessels 2026) reproduce this 0-centered definition verbatim, while Fabbri (2006), facing a genuinely non-self-adjoint operator, introduced explicit remainder terms in the subsolution inequality rather than shifted centers.

Problem Statement

Setting. Let $H$ be a real separable Hilbert space with inner product $⟨·,·⟩$ and norm $‖·‖$, and let $A : D(A) ⊂ H → H$ be a linear, densely defined, maximal monotone operator, not assumed self-adjoint (monotone: $⟨Ax, x⟩ ≥ 0$ for all $x ∈ D(A)$; maximal monotone: $A$ admits no proper monotone linear extension; $A^*$ is the adjoint of $A$, a closed operator with its own dense domain $D(A^*)$, which differs from $D(A)$ in general when $A$ is not self-adjoint). Assume the standing hypotheses of Crandall-Lions Part IV: there exist a constant $C_0$ and a bounded, positive, self-adjoint operator $B$ on $H$ (positive: $⟨Bx, x⟩ ≥ 0$ for all $x ∈ H$) such that $A^*B$ is bounded and the weak B-condition

$$\langle (A^*B + C_0B)x, x\rangle \ge 0 \quad \text{for every } x \in H$$

is satisfied.

Test functions. Let $Ω ⊂ H$ be open and let $u : Ω → (−∞, ∞]$ be weakly sequentially upper semicontinuous, i.e., $\limsup_{n→∞} u(x_n) ≤ u(x)$ whenever $x_n ⇀ x$ weakly in $H$. Consider the Hamilton-Jacobi equation with unbounded linear term

$$\langle Ax, \nabla u\rangle + F(x, u, \nabla u) = 0 \quad \text{in } \Omega$$

where $F : H × ℝ × H → ℝ$ is continuous (Part IV treats this form, the variant with an additional zeroth-order term — in which $u(x̄)$ is added to the left-hand side of the inequality below — and the evolution equations, all with the same test-function mechanism; the centering question is common to all these forms). A Crandall-Lions test pair is a pair $(φ, g)$ consisting of a good part $φ : H → ℝ$ that is weakly sequentially lower semicontinuous, of class $C^1$ with continuous gradient $∇φ$ taking values in $D(A^*)$, and such that $A^*∇φ$ is continuous on $H$; and a bad part $g : H → ℝ$ of class $C^1$ that is radial about the origin and nondecreasing in the radius: $g(x) = g̃(‖x‖)$ for a $C^1$ function $g̃ : [0, ∞) → ℝ$ with $g̃′ ≥ 0$.

The two formulations. In the 0-formulation (Part IV, Definition 1.1), $u$ is a viscosity subsolution in $Ω$ if: for every test pair $(φ, g)$ and every point $x̄ ∈ Ω$ at which $u − (φ + g)$ attains a local maximum, with locality taken in the norm topology of $H$, the inequality

$$\langle x̄, A^*\nablaφ(x̄)\rangle + F(x̄, u(x̄), \nablaφ(x̄) + \nabla g(x̄)) \le 0$$

holds. Here the unbounded term is evaluated through the good part only, via $⟨Ax̄, ∇(φ+g)(x̄)⟩ := ⟨x̄, A^*∇φ(x̄)⟩$, while the full gradient $∇φ + ∇g$ is kept inside $F$; the bad part's would-be pairing $g̃′(‖x̄‖)⟨Ax̄, x̄⟩/‖x̄‖$ (defined at least when $x̄ ∈ D(A)$ and $x̄ ≠ 0$) is nonnegative by monotonicity, which is what licenses discarding it. In the $x_0$-formulation (Part IV, Definition 1.1'), the definition is the same except that the radial bad part may be centered at an arbitrary point $x_0 ∈ D(A)$: $g(x) = g̃(‖x − x_0‖)$ with $g̃′ ≥ 0$, so that the bad part's would-be pairing at a test point $x̄$ is $g̃′(‖x̄ − x_0‖)⟨Ax̄, x̄ − x_0⟩/‖x̄ − x_0‖$, whose monotonicity control passes through the center term $⟨Ax̄, x_0⟩$: if $x_0 ∈ D(A^*)$ then $⟨Ax̄, x_0⟩ = ⟨x̄, A^*x_0⟩$, a function of $x̄$ continuous on all of $H$, and if $A$ is self-adjoint then likewise $⟨Ax̄, x_0⟩ = ⟨x̄, Ax_0⟩$; but for $x_0 ∈ D(A) ∖ D(A^*)$ the center term is well defined for $x̄ ∈ D(A)$ (both arguments lie in $H$) and is in general discontinuous and uncontrolled as a function of $x̄$. Supersolutions are defined by the mirror conditions, exactly as in Part IV, Definition 1.1: test functions touch $u$ from below (local minima of $u − ψ$) and all inequalities are reversed, the monotonicity of the bad part being used with the correspondingly reversed sign.

Problem. Prove or disprove: for every operator $A$ as above satisfying the standing hypotheses with some admissible $B$ and $C_0$, and every weakly sequentially upper (respectively lower) semicontinuous $u$, $u$ is a viscosity subsolution (respectively supersolution) in the 0-formulation if and only if it is one in the $x_0$-formulation, in which the radial bad parts may be centered at arbitrary points $x_0 ∈ D(A)$. Known partial results: the equivalence holds for centers $x_0 ∈ D(A^*)$ for every maximal monotone $A$ (Part IV, Appendix, Proposition A.1), and for centers $x_0 ∈ D(A)$ when $A$ is self-adjoint; Part V records the stronger, $D(A)$-centered form of the subsolution inequality (its (3.8)) as known only in the self-adjoint case. The open core is the case of centers $x_0 ∈ D(A) ∖ D(A^*)$ with $A$ not self-adjoint, where the center term $⟨Ax, x_0⟩$ is well defined but generally discontinuous in $x$.

Scope note. Nonlinear unbounded operators and non-separable Hilbert spaces are outside this task: the nonlinear-operator extension is treated in Part VI (1994), and extensions to other settings proceed through the different definitions of Tataru and others. A resolution must work within the exact Crandall-Lions classes above; adding hypotheses (for example, assuming $A$ self-adjoint, restricting centers to $D(A^*)$, or strengthening the solution class to B-continuous functions) would answer a different, easier question.

References

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