Partial differential equations
16 families · 11 verified by lean
This page gathers the 16 result families that the openai/math repository catalogues under Partial differential equations. Each claim comes from a manuscript; 11 of them have a Lean formalization checked by software, and the other 5 are still to be verified. Open any card to read a plain-language explainer, the original statement and its sources.
Global smoothness for relativistic Vlasov–Maxwell
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.
Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L1 and satisfy exact local conservation of mass, momentum and kinetic energy.
Kinetic limits and fluctuations over the Boltzmann lifespan
Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.
Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.
The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds
Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a arc, a regular crack tip, or three arcs meeting at ; only finitely many global connected components meet any compact interior region.
The critical dimension for the one-phase Bernoulli problem
Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most in higher dimensions.
The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every homeomorphism between arbitrary bounded domains in ℝ3, for , is a strong limit of smooth diffeomorphisms onto the same target.
The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, and real A, B, the system , has no positive entire solution when , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and .
Stable blowup for the defocusing Schrödinger equation
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in , with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.
Global uniqueness in smooth isotropic elasticity
Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.
Nonattainment of the three-marginal Coulomb Monge problem
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.
Sharp one-third stability of Brenier maps
For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.
De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution of that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions as constant wells or planar transitions, without an energy-growth assumption.
Universal computation in forced Navier–Stokes flows
Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.
Interior regularity for infinity-harmonic functions
Proves uniform interior regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.