Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations

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. (Lean)

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,α arc, a regular crack tip, or three arcs meeting at 120°; only finitely many global connected components meet any compact interior region.

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 n-7 in higher dimensions. (Lean)

Resolves the three-dimensional Ball–Evans approximation problem: every W^1,p homeomorphism between arbitrary bounded domains in ℝ3, for 1 ≤ p < ∞, is a strong W^1,p limit of smooth diffeomorphisms onto the same target.

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. (Lean)

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,q > 0 and real A, B, the system -Δ u=|x|^A v^p, -Δ v=|x|^B u^q has no positive entire solution when (n+A)/(p+1)+(n+B)/(q+1) > n-2, 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 A,B > -2. (Lean)

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. (Lean)

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy ‖T_μ-T_ν‖_L²(ρ) ≤ C W₂(μ,ν)^1/3 uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound. (Lean)

Proves uniform interior C^1,α_d 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.