Unit Gradient Functions in 3D Analysis

• Functions with unit gradient are Lipschitz functions defined on bounded, connected domains in ℝ³ that satisfy |∇w| = 1 almost everywhere, forming the basis of relaxed conformality systems.
• This framework generalizes the classical 2D Cauchy–Riemann equations to three dimensions by employing harmonic conjugate pairs whose gradients are mutually orthogonal, derived via a variational approach.
• The theory has significant implications in potential theory, geometric analysis, and inverse problems such as Calderón’s, indicating potential non-uniqueness in boundary measurements.

A function with unit gradient refers to a Lipschitz function $w$ defined on a bounded, connected Lipschitz domain $\Omega \subset \mathbb{R}^3$ such that $|\nabla w(x)| \equiv 1$ almost everywhere in $\Omega$ (Pedregal, 13 Feb 2025). Such functions are central in the study of relaxed conformality systems in three dimensions, particularly in connection to the existence of harmonic conjugate pairs (functions $u,v$) whose gradients, together with $\nabla w$, form an orthonormal frame at almost every point. This relaxation generalizes the Cauchy–Riemann equations to three dimensions and has deep implications for potential theory, geometric analysis, and inverse problems.

1. Relaxing Conformal Constraints in $\mathbb{R}^3$

The classical Cauchy–Riemann equations in the plane are associated with the conformality condition encoded as $D u \cdot (D u)^T = (\det D u) I_2$. In dimensions $N \geq 3$, the analogous system $V u \cdot V u^T = (\det V u)^{2/N} I_N$ admits only degenerate smooth solutions (see Liouville’s theorem). Pedregal introduced a relaxation in three dimensions: instead of full conformality, require that a Lipschitz function $w$ has $|\nabla w(x)| = 1$ almost everywhere, and seek two harmonic functions $u,v$ such that $\nabla u, \nabla v, \nabla w$ are mutually orthogonal. The precise system is given by imposing

• $\nabla u \cdot \nabla v = 0$
• $|\nabla u| = 1$
• $\Delta u = 0,\ \Delta v = 0$ This system can be equivalently encoded as

$$\nabla u = \nabla v \times \nabla w, \qquad \nabla v = \nabla w \times \nabla u.$$

Pairs $(u,v)$ satisfying these are termed conjugate harmonic pairs with respect to $w$ (Pedregal, 13 Feb 2025).

2. Existence of Conjugate Harmonic Pairs with Unit Gradient

Pedregal’s main theorem establishes that for any bounded, connected Lipschitz domain $\Omega$ and any $w \in W^{1,\infty}(\Omega)$ with $|\nabla w| \equiv 1$ almost everywhere, there exists a nontrivial pair $(u,v) \in H^1(\Omega; \mathbb{R}^2)$ (i.e., neither $u$ nor $v$ is constant) such that

• $\Delta u = 0$, $\Delta v = 0$ in $\Omega$
• $\nabla u = \nabla v \times \nabla w$, $\nabla v = \nabla w \times \nabla u$ almost everywhere In this setting, $|\nabla u| = |\nabla v|$ equals the Jacobian determinant $\det[\nabla u, \nabla v, \nabla w]$. This yields infinitely many nontrivial harmonic conjugates of one another under the unit-gradient constraint, contrasting sharply with the rigidity of the classical conformality system in higher dimensions (Pedregal, 13 Feb 2025).

3. Variational Formulation and Proof Strategy

The existence proof employs a variational approach inspired by the two-dimensional theory. The energy functional is defined as

$$J(u,v) = \frac{1}{2}\int_\Omega (|\nabla u|^2 + |\nabla v|^2)\,dx$$

subject to the Jacobian-type constraint

$$\int_\Omega \det[\nabla u, \nabla v, \nabla w]\,dx = c \neq 0$$

Step 1 (Existence): Using direct methods in the calculus of variations, uniform estimates, weak compactness, and the Piola identity $\operatorname{div}(\nabla v \times \nabla w) = 0$, the existence of minimizers is proven for any prescribed nonzero $c$.

Step 2 (Euler–Lagrange System): First variations lead to the condition that, for some Lagrange multiplier $\lambda$,

$$\int_\Omega [(\nabla u - \lambda (\nabla v \times \nabla w)) \cdot \nabla \phi + (\nabla v - \lambda (\nabla w \times \nabla u)) \cdot \nabla \psi]\,dx = 0$$

for all test functions $\phi, \psi \in H^1(\Omega)$. Substituting test fields yields $\lambda = 1$ and recovers the target system. Elliptic regularity then implies $\Delta u = \Delta v = 0$ (Pedregal, 13 Feb 2025).

4. Boundary Conditions for Harmonic Conjugate Pairs

The variational setting prescribes natural boundary conditions, which for $(u,v)$ are:

• $(\nabla u + \nabla v \times \nabla w) \cdot n = 0$
• $(\nabla v - \nabla w \times \nabla u) \cdot n = 0$ on $\partial\Omega$ Alternatively, the condition $(\nabla v \times \nabla w)\cdot n = 0$ can be imposed, representing a "mixed" boundary condition analogous to prescribing tangential derivatives, combining Dirichlet and Neumann aspects for $v$. The trial space for the variational problem is:

$$H_w(\Omega) = \{ v \in H^1(\Omega):\ (\nabla v \times \nabla w) \cdot n = 0\ \text{on}\ \partial\Omega \}$$

On this space, $u$ is uniquely recoverable (up to constants) via a scalar variational problem, and analogously, $v$ is determined given $u$ (Pedregal, 13 Feb 2025).

5. Explicit Constructions and Examples

Explicit examples illustrate the framework:

• Taking $w(x_1,x_2,x_3) = x_3 + 2$ so that $\nabla w = (0,0,1)$, the system reduces to classical 2D Cauchy–Riemann equations in $(x_1,x_2)$. For example, choosing $u(x_1,x_2,x_3) = x_1^2 - x_2^2$, $v(x_1,x_2,x_3) = 2x_1 x_2$, both extended independently of $x_3$, yields mutually orthogonal unit gradients.
• For the unit ball $B^3$ and $w(x) = \sqrt{x_1^2 + x_2^2 + (x_3 + 2)^2}$, $\nabla w$ is the unit radial vector centered at $(0,0,-2)$, and harmonic pairs $(u,v)$ can be constructed axisymmetrically or via variational means (Pedregal, 13 Feb 2025).

6. Implications for Calderón’s Inverse Problem

The Calderón inverse conductivity problem seeks to determine an unknown conductivity $\gamma$ from the Dirichlet-to-Neumann map $$\Lambda_\gamma: u \mapsto \gamma \partial_n u|_{\partial\Omega}$$, where $u$ solves $\operatorname{div}(\gamma \nabla u) = 0$. In two dimensions, $\Lambda_\gamma$ uniquely determines $\gamma$ (Astala–Päivärinta). In three dimensions, uniqueness is unresolved and indeed conjectured to fail at low regularity. Pedregal's construction shows that for any $\gamma(x)>0$ and any unit-gradient $w$, one can construct nontrivial pairs $(u,v)$ with

• $\operatorname{div}(\gamma \nabla u) = 0$
• $\operatorname{div}(\gamma \nabla v) = 0$
• $\gamma \nabla u = \nabla v \times \nabla w$
• $\nabla v = \nabla w \times \nabla u$ Here, $w$ is "invisible" to boundary measurements, introducing a gauge freedom and suggesting that in 3D, the Dirichlet-to-Neumann map may not distinguish different internal structures. This opens avenues for potential non-uniqueness counterexamples in three-dimensional Calderón-type problems (Pedregal, 13 Feb 2025).

7. Structural Role of the Unit Gradient Constraint

The requirement $|\nabla u| = 1$ enforces that $u$ is, locally, a distance-type function: its level sets form parallel surfaces at unit spacing. This "calibration" constraint (Editor's term) is ubiquitous in differential geometry, often simplifying PDEs by linearizing curvature. In the context of potential theory, it bridges the Laplace equation $\Delta u = 0$ with the eikonal-type equation $|\nabla u| = 1$, connecting the domains of wave propagation and harmonic functions. The orthogonality requirements among $\nabla u, \nabla v, \nabla w$ replace the full conformality of classical systems with a less rigid but structurally rich configuration, admitting infinitely many nontrivial solutions and providing robust analytic and variational tools for their study (Pedregal, 13 Feb 2025).