David Pan
Papers
Forcing Sets for Maximum Principles
Which prescribed data force a maximum principle. One method throughout — find the greatest element, then ask it — carried from polynomials to disconjugate operators, to sequences on a grid, to polyharmonic functions on balls and on arbitrary domains, and back to one dimension. It has a proved edge — with clamped data it works exactly on the domains whose Green function stays nonnegative — and a proved ceiling: every way of splitting data between two endpoints obeys a single exponent law, minimized when the endpoints share equally. The question comes from the 2013 note below.
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Which Endpoint Data Force the Maximum Principle for High-Order Derivatives?
Determines exactly which endpoint data force the maximum principle for functions with a nonnegative nth derivative. The forcing set is a union of two closed convex cones — not a finite union of convex polyhedra once n ≥ 4, though the n = 4 cone is determined completely — and an exact overshoot theorem gives the largest possible excess over max{f(a), f(b)}: 1/4 for n = 3, increasing strictly to e − 1.
Extensions
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Forcing Sets for Maximum Principles of Disconjugate Operators
Shows the forcing theory never depended on polynomials. For any disconjugate operator L in Pólya form annihilating constants, the admissible class still has a pointwise-greatest element, so the forcing set is again a union of two convex cones. The sign-chain proof through the Pólya quasi-derivatives is simpler than the original arguments even for L = Dn, and the sharp constant 2 is revealed as the weight of a generalized trapezoidal rule.
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Which Endpoint Data Force the Maximum Principle for Higher-Order Convex Sequences?
The discrete installment, for sequences with nonnegative nth differences. The sign-chain argument transfers verbatim and the sharp n = 3 constant is the continuous one with no discretization correction. Two things are genuinely discrete: the forcing cone's boundary has a single mechanism, since a critical endpoint on a grid is already an interior tie, and the cone is a finite union of two polyhedral cones for every n and N — so the curved walls of the continuous theory are a pure continuum effect.
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Which Data Force the Maximum Principle for Radial Polyharmonic Functions on a Ball?
Carries the framework into genuinely polyharmonic territory: radial functions with Δmu ≥ 0 on a ball, given center data and boundary value. In t = |x|² the radial Laplacian becomes a degenerate Pólya tower, so the class again has a greatest element and the forcing set is linearly isomorphic to the polynomial cone — the whole geometry transfers. Every datum forces for the biharmonic class, the triharmonic criterion is sharp, and none of it needs Boggio's theorem: disconjugacy of the radial operator stands in for Green-function positivity.
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Forcing Navier Data for Polyharmonic Maximum Principles
Reaches general domains. Navier conditions factor Δmu ≥ 0 into iterated Dirichlet Laplacians, so second-order maximum principles do all the work and no positivity theory of polyharmonic Green functions is needed anywhere: the class has a greatest element for m odd and a least element for m even, the parity phenomenon resurfacing at the PDE level. For m = 3 with constant data the forcing cone has exactly two facets, whose slopes are the torsion ratios sup S/(−T) and inf S/(−T) — apparently new domain functionals.
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Forcing Dirichlet Data for Polyharmonic Maximum Principles
Finds the method's edge, and closes the series. On a smooth bounded domain four things are equivalent: the Dirichlet Green function is nonnegative, the class compares one-signed with its interpolant, some clamped datum forces, and every constant boundary value forces. Where the Green function changes sign — on sufficiently eccentric ellipses — no clamped datum forces at all. That explains in hindsight why the five before it could route around Green-function positivity: with clamped data the avoidance is impossible. On the ball the constant-data cone is the classical cone of nonnegative polynomials, and unlike its Navier counterpart it is independent of the dimension.
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Which Two-Point Hermite Data Force the Maximum Principle for u(2m) ≥ 0?
Returns to one dimension for the clamped beam: which two-point Hermite data — heights and derivatives prescribed independently at both ends — force the maximum principle. The forcing set is two convex cones exchanged by reflection, each isomorphic to the classical cone of nonnegative polynomials, and with balanced data curvature appears at the lowest nontrivial order rather than waiting until the fourth. For the beam itself the whole theory is one inequality: it stays above its lower support exactly when u′(a) ≥ 0 and u′(b) ≤ 3δ + 2√(u′(a)δ). The worst sag is the 2013 note's parabola again, mirrored, and the overshoot ceiling is √e − 1 — exactly half the exponent of the one-sided e − 1.
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Forcing Two-Point Hermite Data: The Full Family of Splits
Unifies the one-sided theory of the first paper with the balanced theory of the one before it. For data split (p, q) between the two endpoints, disconjugacy hands every split a forcing theory, and the geometry turns out to be split-blind: the order decides everything, the split nothing. The ceiling theorem is the payoff — as p/n → λ the worst-case overshoot tends to emax(λ, 1−λ) − 1, which settles the exponent conjecture, makes the previous paper's √e − 1 unconditional, and recovers the first paper's e − 1 as the λ → 1 endpoint. Sharing the data equally between the endpoints is asymptotically the best way to control the class.
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Centrality of the Balanced Two-Point Overshoot
Removes the series' last conditional statement. The exact worst-case overshoot at each finite order rested on the extremal Sm being maximized at the midpoint — checked through m = 24, unproven in general. Three steps settle it: differentiating a binomial tail leaves a single Bernstein monomial, the symmetric terms in S′m cancel exactly, and what survives dominates its mirror monomial by monomial. So the midpoint is the unique maximizer at every order, the overshoot table becomes unconditional, and the approach to √e − 1 is sharp, with deficit √e / (2√(πm)). The exact cancellation is what symmetry buys; the asymmetric case stays open.
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Forcing Two-Point Data for n-Convex Sequences: The Discrete Family of Splits
Completes the matrix — {continuous, discrete} × {all splits} — by discretizing the full family. For n-convex sequences with the first p and last q values prescribed, the plain Lagrange interpolant is the barrier, by a four-line divided-difference argument. Three features of the continuous family survive in exact discrete form: the cardinals carry the same binomial coefficients, the sign criterion forces at every split, and the near-balanced tie holds with a shorter proof than the continuum's. Curvature does not — on every grid and every split the forcing set is two polyhedral cones, so curvature is a continuum effect uniformly across the family.
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A Maximum Principle for High-Order Derivatives
Gives conditions under which a nonnegative nth derivative implies the maximum principle f ≤ max{f(a), f(b)}: a pointwise bound by a modified Taylor polynomial, and the corollary that the principle holds whenever f(b) dominates the Taylor polynomials of f at a of orders 1, …, n−2.