Viscosity Solutions Can’t Pin Down the RHS: A 4D Counterexample

Viscosity Solutions Can’t Pin Down the RHS: A 4D Counterexample

Introduction: When one function can satisfy many “different” equations

Hamilton–Jacobi equations show up everywhere—control theory, optics, even the shape of waves and interfaces. A central idea in the modern theory is that when you pick a continuous “potential” function (u) and a Hamiltonian (H), you should get a single value for what the equation is forcing on the right-hand side. In smooth (classical) settings, this intuition is hard to break: if (u) is differentiable, then (H(Du)) is determined point-by-point.

But new research from the paper on arXiv shows a really surprising pathology in the viscosity-solution world: one fixed continuous function, paired with one fixed smooth autonomous Hamiltonian, can satisfy (H(Du)=c) for every (c) in an entire interval. Even more sharply, the function they build is nowhere differentiable inside the domain—so the usual “differentiate and read off the value” intuition simply cannot kick in.

The main result is constructed in dimension four. They build a continuous function (u\in C(\overline{B_1^4})) and a smooth Hamiltonian (H:\mathbb{R}^4\to[1,2]) such that (u) is a viscosity solution of (H(Du)=c) for every (c\in[1,2]) (and in particular it also solves the cases (c=1) and (c=2)). The upshot: viscosity solutions do not always have “single-valuedness” in higher dimensions—continuity of (H) is not enough to guarantee the equation’s right-hand side is uniquely determined.

Why This Matters

Significance RIGHT NOW: robustness vs. uniqueness is not guaranteed

Viscosity solutions are designed to be robust when differentiability fails. That’s their superpower—and also their danger. The paper pinpoints a subtle failure mode: in higher dimensions, the viscosity definition relies on test functions touching (u) from above or below. If (u) is arranged so that those touches occur in incompatible ways at different points (or don’t occur at all in certain senses), then the viscosity inequalities can “fit” multiple values of (c) simultaneously.

This matters in any workflow where people treat viscosity solutions as a stable, unambiguous substitute for classical PDEs. If your model inference step depends on uniqueness of the “effective forcing level,” this counterexample tells you: you can’t always recover it from (u) and (H) alone.

A scenario you can apply TODAY: inverse problems for Hamilton–Jacobi models

Imagine you’re doing an inverse problem: you observe a macroscopic function (u) (say, a measured “potential” or “cost-to-go” landscape), and you already believe the system is governed by some Hamiltonian-like structure (H). In many idealized derivations, you’d infer the constant forcing (c) from the identity (H(Du)=c).

The pathology here says that in 4D (and via related constructions in other settings), the same observed (u) could be consistent with multiple forcing levels—even a whole continuum—without changing (H). That means that uniqueness assumptions in inverse inference steps need extra checking (convexity, coercivity, additional regularity, or dimension restrictions).

How this builds on previous AI research (and why it’s a good “constraint check”)

In the AI era, it’s common to use neural solvers, neural operators, or differentiable programming to approximate PDE solutions. Those methods often implicitly assume that the PDE residual corresponds to a unique target. This paper is a reality check: even if your solver finds a function that “looks like” it satisfies a PDE in the viscosity sense, the mapping from ((u,H)) to “the right-hand side” might not be unique.

So while AI research may help find candidate solutions, this result warns that you can’t always interpret the fitted parameter uniquely unless your problem satisfies the structural assumptions that restore single-valuedness.

The Main Mechanism: Why viscosity “single-valuedness” can fail

The classical promise: differentiable (u) fixes (H(Du))

In the classical setting, if (u) is differentiable, then (H(Du)) is unambiguous. If two constants (c) and (c') both satisfy (H(Du)=c) and (H(Du)=c'), then (c=c').

So the natural question the paper addresses is:

In the viscosity framework, does the same uniqueness still hold?

The viscosity twist: test functions can appear at different places—or not at the same points

For a continuous function (u), viscosity solutions are defined using (C^1) test functions (\varphi) that touch (u) from above or below at a point (x). Informally:
- If (\varphi) touches (u) from above at (x), then gradients (D\varphi(x)) must satisfy the inequality in the “subsolution” direction.
- If (\varphi) touches (u) from below at (x), then gradients must satisfy the inequality in the “supersolution” direction.

The paper uses standard notation for the first-order superdifferential/subdifferential (D^+u(x)), (D^-u(x)). Here’s the key observation they exploit:

• If a point (x) has no upper test functions (so (D^+u(x)) is empty) and also no lower test functions (so (D^-u(x)) is empty), then both viscosity requirements are vacuous at that point.

In higher dimensions, you can build pathological (u) so that these “test availability” patterns are engineered—meaning the usual logic that forces a unique right-hand side can collapse.

The target pathology: one (u), one (H), many constants (c)

The authors’ construction aims to violate the expected “single-valuedness” property for Hamilton–Jacobi operators. They construct:
- a continuous (u\in C(\overline{B_1^4})),
- a smooth Hamiltonian (H\in C^\infty(\mathbb{R}^4;[1,2])),

such that (u) is a viscosity solution of (H(Du)=c) in (B_1^4) for every (c\in[1,2]). Concretely, this implies:
- (u) solves (H(Du)=1),
- and also solves (H(Du)=2),
- and in fact solves every intermediate (c) as well.

Worse (and crucial for the pathology), (u) is nowhere differentiable in (B_1^4).

From planar chaos to 4D Hamiltonians: the “lifting” idea

First build a nowhere-differentiable planar function with disjoint gradient behavior

A big portion of the proof is an intermediate construction in a planar domain. They start by building a function (F) on a 2D region whose “upper” and “lower” semidifferential graphs behave very differently.

To explain the setup without heavy notation: consider the set of pairs ((x,p)) where (p) can occur as a gradient of a test function touching (F) from above or from below. The authors manage to ensure that:
- the “upper-touch gradient graph” and
- the “lower-touch gradient graph”
are disjoint in the right augmented sense, and are also relatively closed.

Crucially, they show the disjointness can be strong enough that you can later separate these sets with a smooth Hamiltonian.

Why “disjoint gradients” alone isn’t enough—and how they fix it

A subtle issue appears: if you only ensure that two sets of gradient vectors (p) don’t overlap, that might still fail to force anything meaningful about where the touching occurs. The paper deals with this by using a lifting trick:

• They add extra spatial variables so that when test functions touch in different locations, the gradient encodes where the contact happened.
• That prevents the projection overlap problem.

Once the authors make the “contact location” information part of the gradient record, the disjointness becomes usable.

Turning the planar function into a 4D solution

After constructing the planar (F), they create a higher-dimensional (u) (their (U), then renamed appropriately) by a controlled “lifting” formula. The intuition is:
- you keep (F)’s pathological upper/lower structure,
- but you embed it into a higher-dimensional geometry where a smooth Hamiltonian can be manufactured to separate the two gradient-contact patterns.

Then they choose (H) as a smooth “separation function” for the two disjoint sets associated with upper vs. lower touching. Because the sets are relatively closed and disjoint, a smooth separator exists (they rely on a separation argument of the type described in their Lemma/Proposition setup).

This is how one fixed (H) can be made to respond correctly to viscosity inequalities for many different constants (c).

A closer look at the construction logic (without drowning in formulas)

Step 1: Start from a cusp-y base function

In the planar part, they begin with a 2-periodic function (w) that’s smooth except at cusps, with square-root type singular behavior. They list key properties:
- it’s continuous and (2)-periodic,
- smooth on intervals ((j,j+1)),
- bounded by (|w|\le \pi/4),
- with (w(0)=\pi/4) and (w(1)=-\pi/4),
- and its derivative changes sign in each unit interval in a controlled way.

The role of (w) is to create sharp local behavior while remaining controlled enough for an iterative construction.

Step 2: Iteratively refine with polygon cuts and tiny perturbations

They take an open convex polygon (P) and define perturbations of a current function (g) based on distance to the polygon boundary and rescaled versions of the cusp function (w).

Then comes the iterative “divide and perturb” lemma idea:
- Subdivide the polygon into smaller convex polygons using parallel cuts.
- Perturb (g) to (\tilde g), keeping (\tilde g=g) on the polygon boundary so previously built structure remains intact.
- Arrange things so that on each subpolygon, the upper and lower semidifferentials become “one-sided” in a strong sense: at points of interest, one of (D^+ \tilde g) or (D^- \tilde g) becomes empty.

This is how they engineer nowhere differentiability: if both upper and lower differentiability would force matching gradients, but the construction makes one side disappear in different regions, differentiability can’t happen anywhere.

Step 3: Take the limit function with stable pathological structure

They repeat this refinement indefinitely. Because the perturbations get smaller fast enough, the sequence ({f_n}) converges uniformly to a continuous limit (f).

Their proof includes a careful dichotomy for points:
- At points lying on the “infinite skeleton” of created cuts, exactly one semidifferential graph remains nonempty, and the other collapses.
- Off that skeleton, both semidifferentials behave in a controlled way—but in a way that still prevents differentiability.

This gives the planar (F) they need.

How the Hamiltonian is finally chosen to support every (c\in[1,2])

Once the authors have the planar (F) and the lifted 4D structure (U), the remaining job is to build a smooth Hamiltonian (H) so that viscosity inequalities match (H(Du)=c) for all (c) in an interval.

The separator viewpoint: two disjoint graphs → one Hamiltonian

Think of the upper- and lower-touch regimes as creating two disjoint closed sets in an appropriate space of ((\text{contact position}, \text{gradient}))-type variables. Once you have two disjoint closed sets, a standard functional-analytic idea says you can craft a smooth function that:
- takes values in ([1,2]),
- assigns one range of values to one set,
- assigns another range to the other set,
- and does so smoothly.

That’s what they implement: they choose (H\in C^\infty(\mathbb{R}^4;[1,2])) as a smooth separation function for those sets.

Why this supports an entire interval of constants

The final cleverness is that viscosity solutions don’t ask for a pointwise identity (H(Du)=c) in the classical sense. Instead, they use inequalities tested against touching functions.

Because the construction ensures that at each point, the “available” touching information corresponds in a controlled way to one of the two gradient sets, the inequalities can be made compatible with multiple choices of (c). In their setting, compatibility holds for every (c\in[1,2]).

So the single-valuedness intuition fails dramatically: the fixed pair ((u,H)) doesn’t determine a unique forcing level.

Comparisons across dimensions and autonomous vs. non-autonomous settings

The paper also discusses how the phenomenon changes with dimension and whether the Hamiltonian depends on time/position.

They also highlight that the dimension 4 claim is “optimal” in the sense they explain: you can’t just start with a nowhere-differentiable 1D solution and lift it to get a 2D autonomous example (their argument uses monotonicity considerations and the behavior of test functions at dense maxima/minima).

Key Takeaways

• Viscosity solutions can be wildly non-unique: in 4D, a single continuous nowhere-differentiable function (u) with a single smooth autonomous Hamiltonian (H) can satisfy (H(Du)=c) for every (c\in[1,2]).
• Continuity of (H) alone doesn’t restore uniqueness. The expected “single-valuedness” property fails because viscosity definitions depend on where and how test functions touch the solution.
• The construction is built in layers:
  1) create a planar nowhere-differentiable function with carefully engineered disjoint upper/lower gradient behavior,
  2) “lift” it into 4D so the gradient remembers the contact location,
  3) choose (H) as a smooth separator for two disjoint sets.
• Practical implication for inverse problems and parameter fitting: if your model assumes (u) and (H) determine the forcing level (c), this paper is a warning that in higher-dimensional viscosity settings, that inference can be non-unique.
• Dimension matters: the autonomous pathology is established in dimension four, while whether it can occur in dimensions two or three (for the autonomous case) remains an open question in the paper.

If you want, I can also rewrite the core idea (“upper/lower test availability + separation via a Hamiltonian”) into an intuitive diagram-style narrative that doesn’t use any of the heavy notation—just the geometric intuition.