Local Potential Problems: A Survey

• Local potential problems are a family of formulations where global quantities are derived from local constraints or responses, as seen in obstacle problems and distributed graph algorithms.
• They feature diverse applications ranging from nonlocal interaction energies and disordered system optimization to mixed local–nonlocal elliptic PDEs and conceptual DFT challenges.
• Methodologies include localized variational techniques, local error estimations in spin glasses, and controlled density-dependent interactions in mesoscopic and superconducting models.

“Local potential problems” is not a single standardized term; on arXiv it denotes several formally distinct research programs that share a local variational or local equilibrium structure. In one line of work, local minimizers of nonlocal interaction energies induce classical or fractional obstacle problems for the interaction potential $W * \mu$. In another, the term refers to locally checkable optimization or equilibrium conditions on graphs, local ground-state prediction tasks in disordered systems, or localized value functions in minimax optimization. It also appears in mixed local–nonlocal elliptic equations with Hardy-type singular potentials, in coarse-grained molecular modeling through local-density-dependent interaction laws, in nonequilibrium superconductivity through the local pair potential $\Delta(\mathbf r,t)$, and in conceptual density functional theory through unsuccessful attempts to define a local hardness from $\delta\mu/\delta n(\mathbf r)$ (Carrillo et al., 2014, Shen et al., 5 May 2025, Balliu et al., 16 Jul 2025, Biagi et al., 2024, Faure et al., 2013, Gal, 2011). This suggests that the expression is best understood as a family of locality-based formulations rather than a single canonical object.

1. Core meanings and recurring formal structure

Across these literatures, the phrase designates a problem in which a global object is reconstructed from local data, local constraints, or local response. Typical representatives include the interaction potential $\psi=W*\mu$ near points of $\operatorname{supp}\mu$, the local cost $c_v$ and global potential $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$ in distributed graph problems, the local single-bond error $\mathcal E_{ij}$ in spin glasses, and localized minimax value functions $V_\tau(x)$ or $V_{\epsilon_0}(x)$ under coupled constraints (Balliu et al., 16 Jul 2025, Shen et al., 5 May 2025, Ma et al., 4 Oct 2025).

Domain	Local object	Representative formulation
Interaction energies	$\Delta(\mathbf r,t)$0	Local obstacle problem
Distributed optimization	$\Delta(\mathbf r,t)$1	LOP / GLOP
Disordered systems	$\Delta(\mathbf r,t)$2, $\Delta(\mathbf r,t)$3	Local hardness
Mixed PDE	Hardy potential, Riesz/Wolff potentials	Mixed local–nonlocal elliptic problem
Mesoscopic modeling	$\Delta(\mathbf r,t)$4, $\Delta(\mathbf r,t)$5	Local-density-dependent potential
DFT	$\Delta(\mathbf r,t)$6, $\Delta(\mathbf r,t)$7	Local hardness / local chemical potential

A second recurring feature is that “local” rarely means merely “small scale.” In the obstacle-problem setting, locality arises because $\Delta(\mathbf r,t)$8-minimality permits only small spatial perturbations. In distributed graph problems, locality is the constant-radius checkability of a labeling. In disordered systems, local solvers are tested against global critical thresholds and avalanches. In mixed elliptic problems, local singular potentials coexist with nonlocal diffusions, so local behavior is still influenced by tails. In conceptual DFT, the failure of a local hardness definition is traced precisely to the asymptotic behavior of the universal functional, showing that a purportedly local derivative can be controlled by behavior “at infinity” (Carrillo et al., 2014, Balliu et al., 16 Jul 2025, Shen et al., 5 May 2025, Biagi et al., 2024, Gal, 2011).

2. Interaction energies as local obstacle problems

In the interaction-energy setting, the basic functional is

$\Delta(\mathbf r,t)$9

for $\delta\mu/\delta n(\mathbf r)$0, with $\delta\mu/\delta n(\mathbf r)$1 nonnegative, lower semicontinuous, and locally integrable. Local minimality is metric, defined with the $\delta\mu/\delta n(\mathbf r)$2 optimal transport distance, and for an $\delta\mu/\delta n(\mathbf r)$3-local minimizer the associated interaction potential

$\delta\mu/\delta n(\mathbf r)$4

satisfies the local Euler–Lagrange inequality

$\delta\mu/\delta n(\mathbf r)$5

for every $\delta\mu/\delta n(\mathbf r)$6. After continuity is established, this becomes pointwise and strengthens to local flatness on the support: $\delta\mu/\delta n(\mathbf r)$7 The local potential problem is then to identify $\delta\mu/\delta n(\mathbf r)$8 as a solution of a unilateral PDE with coincidence set $\delta\mu/\delta n(\mathbf r)$9 (Carrillo et al., 2014).

For Newtonian repulsion, the decomposition $\psi=W*\mu$0 with $\psi=W*\mu$1 yields

$\psi=W*\mu$2

Fixing $\psi=W*\mu$3 and writing $\psi=W*\mu$4, the potential solves in $\psi=W*\mu$5 the local obstacle problem

$\psi=W*\mu$6

with obstacle $\psi=W*\mu$7 and $\psi=W*\mu$8. For Riesz-type repulsion $\psi=W*\mu$9 with $\operatorname{supp}\mu$0, the same structure becomes fractional: $\operatorname{supp}\mu$1 and $\operatorname{supp}\mu$2 solves a local fractional obstacle problem in which the complement condition holds in $\operatorname{supp}\mu$3 and the exterior datum is prescribed on $\operatorname{supp}\mu$4 (Carrillo et al., 2014).

The regularity theory is dictated by the repulsion strength at the origin. Under Newtonian repulsion, compactly supported $\operatorname{supp}\mu$5-minimizers satisfy $\operatorname{supp}\mu$6 and $\operatorname{supp}\mu$7 with $\operatorname{supp}\mu$8. If $\operatorname{supp}\mu$9, then $c_v$0, and if $c_v$1 near $c_v$2, then $c_v$3 is a set of locally finite perimeter. For more singular-than-Newtonian repulsion, $c_v$4 for every $c_v$5 and $c_v$6 for every $c_v$7. The coincidence set of the obstacle problem identifies, up to negligible sets, with $c_v$8, so free-boundary regularity transfers to the geometry of minimizers (Carrillo et al., 2014).

3. Landscape formulations: local hardness, overlap potentials, and calm local minimax

In disordered spin systems, “local potential problems” are studied through exact local optimization on subsystems. The paper on local optimization in EA and SK spin glasses introduces the local single-bond solver: one selects a target bond $c_v$9, cuts out a finite neighborhood, computes the exact subsystem ground state, and compares $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$0 with the full-system ground-state bond product. The disorder-averaged error rate is

$$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$1

This error decays as a power law in subsystem size for the 2D and 3D EA models,

$$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$2

with fitted parameters $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$3 in 2D and $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$4 in 3D. The key organizing notion is local hardness: how large a local subsystem must be so that a local solver predicts a local observable with a prescribed small average error. That hardness is controlled not by $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$5 versus NP-hard global complexity, but by bond-specific critical thresholds $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$6, gapless avalanche-like excitations, and zero-energy droplets. The local error obeys a stretched-exponential relation in distance to criticality,

$$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$7

and changes in critical thresholds decay algebraically with distance,

$$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$8

with $$\operatorname{Pot}(G,\ell)=\sum_{v\in V}\phi_\ell(G_r(v),v)$$9 in 2D and $\mathcal E_{ij}$0 in 3D (Shen et al., 5 May 2025).

A related landscape formulation appears in the overlap analysis of shortest paths and shortest path trees. There the Franz–Parisi potential is used as a constrained free-energy profile over an overlap parameter $\mathcal E_{ij}$1. For shortest paths, the overlap-gap property and the Franz–Parisi potential both indicate that local search fails in the path landscape, whereas for shortest path trees they indicate that local search should succeed. The tree Gibbs measure is

$\mathcal E_{ij}$2

and the Franz–Parisi potential $\mathcal E_{ij}$3 is defined by constraining the overlap with a typical Gibbs sample. In the tree formulation, the overlap structure is continuous enough that there is no ensemble overlap-gap property, while in the path formulation local search encounters a barrier. Li and Schramm’s earlier lower bound against stable algorithms for shortest paths is thereby reinterpreted as a statement about one unfavorable landscape among two polynomial-time-equivalent formulations (Koehler et al., 24 Nov 2025).

In minimax optimization with coupled constraints, the same local-potential language is used for a localized value function around a candidate equilibrium. The basic problem is

$\mathcal E_{ij}$4

with $\mathcal E_{ij}$5 depending on $\mathcal E_{ij}$6. A calm local minimax point $\mathcal E_{ij}$7 is defined by the existence of a radius function $\mathcal E_{ij}$8 with $\mathcal E_{ij}$9 such that, for all small $V_\tau(x)$0,

$V_\tau(x)$1

This is equivalent to local optimality of a localized value function $V_\tau(x)$2 and inner calmness of the localized argmax map $V_\tau(x)$3. The resulting first- and second-order conditions are stated in terms of tangent cones, graphical derivatives $V_\tau(x)$4, and second subderivatives, and they specialize to KKT-type conditions under MSCQ, RS, MFCQ, RCRCQ, or affine/polyhedral assumptions (Ma et al., 4 Oct 2025).

4. Distributed graph-theoretic local potential problems

In distributed graph algorithms, a local potential problem is formalized as a locally checkable labeling endowed with a nonnegative local potential

$V_\tau(x)$5

and a global potential

$V_\tau(x)$6

A labeling is locally optimal when no node can change only its own output and incident half-edges in a way that strictly decreases its local cost and the potential in its radius-$V_\tau(x)$7 neighborhood. This is the LOP framework; GLOPs extend it by allowing a node to relabel all nodes in its radius-$V_\tau(x)$8 neighborhood, and every GLOP reduces in $V_\tau(x)$9 rounds to an LOP. The main algorithmic result is that on bounded-degree graphs every LOP admits a randomized LOCAL algorithm in $V_{\epsilon_0}(x)$0 rounds w.h.p., and hence every local potential problem in this sense has deterministic and randomized complexity $V_{\epsilon_0}(x)$1. In particular, the deterministic complexity of locally optimal cut is settled to $V_{\epsilon_0}(x)$2 (Balliu et al., 16 Jul 2025).

A complementary classification is available on directed cycles for general local optimization problems of min-sum, max-sum, min-max, or max-min type. An opt LCL is specified by $V_{\epsilon_0}(x)$3, where $V_{\epsilon_0}(x)$4 is a local cost or utility on radius-$V_{\epsilon_0}(x)$5 neighborhoods and the global objective aggregates these local quantities by $V_{\epsilon_0}(x)$6, $V_{\epsilon_0}(x)$7, or $V_{\epsilon_0}(x)$8. For every such problem and every constant approximation ratio $V_{\epsilon_0}(x)$9, the complexity on directed cycles falls into exactly one of four classes: $\Delta(\mathbf r,t)$00 deterministic and randomized; $\Delta(\mathbf r,t)$01 deterministic and $\Delta(\mathbf r,t)$02 randomized; $\Delta(\mathbf r,t)$03 in both models; or $\Delta(\mathbf r,t)$04 in both models. Moreover, the complexity class and an asymptotically optimal algorithm can be found automatically by a centralized meta-algorithm (Boudier et al., 13 Feb 2026).

The broader LCL landscape shows that some local problems do separate stronger models, but trees are unusually rigid. One constructed LCL requires $\Delta(\mathbf r,t)$05 rounds with private randomness yet drops to $\Delta(\mathbf r,t)$06 with shared randomness, resolving the question of whether shared randomness ever helps with LCLs (Balliu et al., 2024). Another LCL, iterated GHZ, is solvable in one quantum-LOCAL round but requires $\Delta(\mathbf r,t)$07 rounds classically, giving the first super-constant distributed quantum advantage for a local problem (Balliu et al., 2024). By contrast, in trees many LCL classes have the same locality in deterministic LOCAL and randomized online-LOCAL; this yields corollaries that there is no distributed quantum advantage for those classes on trees, and that problems global in deterministic LOCAL remain global across all stronger models captured by randomized online-LOCAL (Dhar et al., 2024).

5. Mixed local–nonlocal elliptic problems with singular potentials and measure data

In PDE, one major meaning of local potential problems concerns mixed local–nonlocal operators with Hardy-type singular terms. A model case is

$\Delta(\mathbf r,t)$08

where $\Delta(\mathbf r,t)$09 is bounded, $\Delta(\mathbf r,t)$10, and $\Delta(\mathbf r,t)$11. The natural energy space is

$\Delta(\mathbf r,t)$12

The decisive Hardy fact is that for the mixed norm the optimal Hardy constant remains the classical local one,

$\Delta(\mathbf r,t)$13

and this value is not attained. Consequently, the critical potential is still $\Delta(\mathbf r,t)$14, not $\Delta(\mathbf r,t)$15, and the solvability theory closely parallels the purely local Hardy problem. For $\Delta(\mathbf r,t)$16 with $\Delta(\mathbf r,t)$17 and $\Delta(\mathbf r,t)$18, there is a unique weak solution in $\Delta(\mathbf r,t)$19; for lower summability one obtains distributional or duality solutions, and for $\Delta(\mathbf r,t)$20 solvability is characterized by the condition $\Delta(\mathbf r,t)$21, where $\Delta(\mathbf r,t)$22 solves the mixed Hardy equation with right-hand side $\Delta(\mathbf r,t)$23 (Biagi et al., 2024).

A second PDE line studies mixed local–nonlocal $\Delta(\mathbf r,t)$24-Laplace type equations with measure data,

$\Delta(\mathbf r,t)$25

with $\Delta(\mathbf r,t)$26 a finite signed Borel measure, $\Delta(\mathbf r,t)$27 a Carathéodory vector field satisfying $\Delta(\mathbf r,t)$28-growth and ellipticity, and $\Delta(\mathbf r,t)$29 merely measurable in the nonlocal term. The nonlinear potential theory is formulated through truncated Wolff and Riesz potentials,

$\Delta(\mathbf r,t)$30

together with a nonlocal tail

$\Delta(\mathbf r,t)$31

The paper proves universal pointwise estimates for $\Delta(\mathbf r,t)$32 and $\Delta(\mathbf r,t)$33 via Riesz and Wolff potentials, introduces a novel fractional maximum function that captures local and nonlocal features simultaneously, and shows that local oscillations of solutions are determined by local energy averages, the tail term, and nonlinear potentials of $\Delta(\mathbf r,t)$34 (Ma et al., 15 Oct 2025).

A third line considers a quasilinear mixed operator with a mixed interpolated Hardy potential,

$\Delta(\mathbf r,t)$35

The mixed Hardy inequality

$\Delta(\mathbf r,t)$36

controls the singular local potential by the mixed energy. The paper establishes a concentration–compactness principle in which concentration defects occur only at the singular point $\Delta(\mathbf r,t)$37, combines it with Ricceri’s variational principle to obtain nontrivial weak solutions for subcritical nonlinearities, and applies the mountain pass theorem to the superlinear case (Aikyn et al., 28 Jun 2026).

6. Mesoscopic, superconducting, and conceptual-DFT formulations

In coarse-grained molecular modeling, a local potential problem arises when the pair interaction must depend on the local environment in order to encode mesoparticle compressibility. The basic ansatz is

$\Delta(\mathbf r,t)$38

where $\Delta(\mathbf r,t)$39 is a local volume associated with particle $\Delta(\mathbf r,t)$40. Two choices are compared: spherical local density $\Delta(\mathbf r,t)$41, which depends on a kernel $\Delta(\mathbf r,t)$42 and a cutoff $\Delta(\mathbf r,t)$43, and Voronoi-based density $\Delta(\mathbf r,t)$44, where $\Delta(\mathbf r,t)$45 is the Voronoi cell volume. The Voronoi approach is parameter-free and satisfies $\Delta(\mathbf r,t)$46 exactly. In nitromethane, with one mesoparticle representing one thousand molecules, a quadratic $\Delta(\mathbf r,t)$47 fitted against Hugoniot data reduces the RMS pressure error from $\Delta(\mathbf r,t)$48 GPa for $\Delta(\mathbf r,t)$49 to $\Delta(\mathbf r,t)$50 GPa for $\Delta(\mathbf r,t)$51, and remains accurate on higher-density test states (Faure et al., 2013).

In nonequilibrium superconductivity, the local pair potential is

$\Delta(\mathbf r,t)$52

within the time-dependent Bogoliubov–de Gennes equations. The proposed mechanism is to use a control field $\Delta(\mathbf r,t)$53 to localize Bogoliubov quasiparticle amplitudes near a target point $\Delta(\mathbf r,t)$54, thereby enhancing $\Delta(\mathbf r,t)$55 and the associated local effective transition temperature

$\Delta(\mathbf r,t)$56

For a one-dimensional model with $\Delta(\mathbf r,t)$57 and a multi-frequency $\Delta(\mathbf r,t)$58, the paper reports about $\Delta(\mathbf r,t)$59 enhancement of the local pair potential and local $\Delta(\mathbf r,t)$60 compared with the initial state. The effect is transient and explicitly constrained by the weak-nonequilibrium regime of the TDBdG equations, pair breaking for $\Delta(\mathbf r,t)$61, decoherence, and phase slips in thin systems (Grigorenko et al., 2011).

Conceptual DFT supplies an explicitly critical perspective. The traditional local hardness is defined by

$\Delta(\mathbf r,t)$62

and the analogous local chemical potential by $\Delta(\mathbf r,t)$63. The analysis shows that the fixed-$\Delta(\mathbf r,t)$64 derivative is only a restricted derivative and is therefore ambiguous; if it is interpreted as a proper constrained derivative, it collapses to a constant value equal to the global hardness $\Delta(\mathbf r,t)$65, and the local chemical potential likewise becomes the constant $\Delta(\mathbf r,t)$66. If the external-potential constraint is dropped, the resulting derivative is tied to the asymptotic value of the second derivative of the universal functional,

$\Delta(\mathbf r,t)$67

which is then undermined by asymptotic divergence. The paper therefore concludes that the traditional approach is incapable of delivering a local hardness indicator and that the parallel local chemical-potential construction fails for the same reason (Gal, 2011).

These physical and chemical usages make clear that “local potential problems” can denote either a constructive modeling strategy or a diagnosis of conceptual failure. In the mesoparticle and superconducting settings, locality is engineered to make the potential responsive to environment or control. In conceptual DFT, the same localization program breaks down because the supposed local quantity is governed by restricted derivatives or asymptotic functional behavior rather than chemically meaningful sitewise response (Faure et al., 2013, Grigorenko et al., 2011, Gal, 2011).