Analysis of PDEs
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Paper 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…
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Paper 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…
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Paper 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…
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Paper Interior C^{1,\alpha} regularity for infinity-harmonic functions
Proves uniform interior C^{1,\alphad} 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…
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Paper De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:\mathbb R^8\to(-1,1) of \Delta u=u^3-u that is strictly increasing in one direction depends on only one linear…
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Paper 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…
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Paper 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 \|T\mu-T\nu\|{L^2(\rho)}\le C W2(\mu,\nu)^{1/3} uniformly over targets in a…
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Paper The planar Mumford–Shah regularity conjecture and local weak-L^{4} 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 C^{1,\alpha} arc, a regular crack tip, or three…
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Paper The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q\gt 0 and real A, B, the system -\Delta u=|x|^A v^p, -\Delta v=|x|^B u^q has no positive entire solution when…
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Paper 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…
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Paper 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…
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Paper Stable blowup for the defocusing Schrödinger equation
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in H^k(\mathbb T^{12}), with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up…
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Paper 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 3\lambda+2\mu\gt 0 on the closure. Neither…
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Paper The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every W^{1,p} homeomorphism between arbitrary bounded domains in ℝ3, for 1\le p\lt \infty, is a strong W^{1,p} limit of smooth diffeomorphisms…
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Paper 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…
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Paper 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…
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