General · EdgepediaOctober 10, 20265 min read

Norman George Meyers

Norman George Meyers (born June 29, 1930, Buffalo, New York) was an American mathematician who specialized in partial differential equations, spent his career at the University of Minnesota, and is best known for the 1963 higher-integrability result for elliptic divergence-form equations now called the Meyers estimate or, jointly with its planar precursor, the Boyarsky–Meyers estimate.¹^•²

Key fact	Detail
Born	June 29, 1930, Buffalo, New York, first of two children of Morris and Ida Ruth, Jewish immigrants from Central and Eastern Europe¹
Training	B.A. in mathematics, University of Buffalo, 1952, on scholarship; Ph.D., Indiana University, 1957, advisor David Gilbarg¹^•³
Career	University of Minnesota, except 1968–69 at Hebrew University, Jerusalem, and 1972–73 at Indiana University Bloomington; retired at age 80¹
Signature result	1963 Lᵖ-estimate for the gradient of solutions of second-order elliptic divergence equations, Annali della Scuola Normale Superiore di Pisa 17 (3), 189–206²
Other major papers	Meyers–Elcrat, Duke Math. J. 42 (1975), 121–136; Meyers–Serrin "H = W," PNAS 51 (1964); quasi-convexity paper, Trans. AMS (1965)⁴^•⁵
Death	Evening of April 30, 2025, age 94, after Parkinson's disease; funeral May 5 at Minneapolis Jewish Cemetery Chapel, Richfield¹

Life and education

Meyers was born in Buffalo. He took his mathematics degree at the University of Buffalo in 1952 on scholarship, then moved to Indiana University, where he completed a Ph.D. in 1957 under David Gilbarg; his dissertation was titled Asymptotic Behaviour of Solutions of Linear Elliptic Differential Equations.¹^•³

His teaching career was spent almost entirely at the University of Minnesota, with two interruptions: the 1968–69 academic year at Hebrew University in Jerusalem and 1972–73 at Indiana University in Bloomington. He retired at age 80. A bibliometric profile also records an IBM affiliation in 1960.¹ His wife Harriet (née Schweller) died in 2015; the couple moved from Minneapolis's Kenwood neighborhood to St Louis Park in 2005, and in June 2022 he moved to The Pillars of Prospect Park assisted living facility. The obituary also remembers him as an exceptional squash player and athlete.¹

Mathematical work

Meyers's field was partial differential equations, with research areas recorded as nonlinear PDEs, harmonic analysis, geometric analysis, and boundary problems.¹ Four results anchor his reputation.

The 1963 estimate. In An Lᵖ-estimate for the gradient of solutions of second order elliptic divergence equations (Annali della Scuola Normale Superiore di Pisa, 3e série, tome 17, no 3, pp. 189–206), Meyers extended a planar result of Boyarskii to n-dimensional elliptic equations of divergence structure, proving the existence of a number Q > 2, depending only on the ellipticity constants and the dimension n, such that the gradient of a solution gains integrability beyond L². The same paper proves an existence and uniqueness theorem for the Dirichlet problem on sufficiently smooth domains.²

The Meyers–Elcrat extension. With Elcrat he carried the higher-integrability idea to higher-order nonlinear systems in Duke Mathematical Journal 42 (1975), pp. 121–136, using a device of Gehring developed for derivatives of quasi-conformal maps; the Meyers–Elcrat system's distinctive feature is that every $W^{m,p}$ -solution has a reverse Hölder exponent q > p. MathSciNet classifies the paper under variational methods for higher-order elliptic equations.⁴^•⁵

H = W. With James B. Serrin he published "H = W" in the Proceedings of the National Academy of Sciences (1964). This two-page paper is the source of the Meyers–Serrin theorem, which states that smooth functions are dense in the Sobolev space $W^{k,p}$ (Ω) for arbitrary domains Ω, so that the weak and strong definitions of Sobolev spaces coincide without any smoothness assumption on the domain.⁹⁹

The Meyers estimate in context

The estimate is a self-improvement property: a weak solution of a divergence-form elliptic equation whose gradient is merely square-integrable automatically has a gradient in Lᵖ for some p > 2. In the form given by a 2025 survey of the first-order approach, there exist p > 2 and a constant C ≥ 0, both depending only on ellipticity and dimension, giving a weak reverse Hölder estimate for the gradient of local L-harmonic functions on every cube; the supremum over all admissible p is denoted m₊(L) and called the Meyers exponent of L.⁷ A typical quantitative form, for a second-order linear elliptic boundary value problem in a bounded strongly Lipschitz domain, is an inequality of the type

$\int_D |\nabla u|^{2+\delta}\,dx \le C \int_D |f|^{2+\delta}\,dx,$

for some δ > 0.⁸

The proof's main tool, as in Boyarskii's argument, is the Calderón–Zygmund inequality for singular integrals, though Meyers noted that its role is largely hidden; the underlying real-variable ideas originate in Gehring's self-improvement property for conformal mappings.²^•⁷ The result sits alongside the De Giorgi–Nash–Moser theory, which had just established Hölder continuity of solutions of elliptic equations (De Giorgi's solution of Hilbert's 19th problem in dimensions larger than two appeared the year before Nash's independent work, and Moser's 1960 new proof in Communications on Pure and Applied Mathematics 13, pp. 457–468, introduced Moser iteration).⁹^•¹⁰ Meyers's higher integrability immediately implies, via the Sobolev lemma, that the solution is Hölder continuous, so it recovers part of that regularity by a different route; the best possible value of the Lebesgue exponent was unknown at the time of writing.² The result also matters against the background of counterexamples to regularity for elliptic systems in dimensions higher than two: full regularity fails for systems, but higher integrability of the gradient survives as a partial positive result.⁹

By the numbers

The Mathematics Genealogy Project lists one doctoral student, David Adams (University of Minnesota, 1969), and two descendants in the mathematical genealogy sense.³ Adams and Xiao's paper dedicated to his 80th birthday appeared in 2012.⁴

Legacy and influence

The Meyers exponent m₊(L) remains a live object of research. A 2025 preprint proves that the exponent p₊(DB) arising from the first-order approach to second-order elliptic boundary value problems coincides with the Meyers exponent; the two capture different aspects of the same regularity, since m₊(L) describes local interior behavior of weak solutions uniformly across all scales while p₊(DB) relates to a global estimate at the boundary.⁷ The higher-integrability property is standardly named the Boyarsky–Meyers estimate in the literature.⁸ The Meyers–Elcrat system continues to be used in singular-set estimates for nonlinear elliptic systems: Adams and Xiao's 2012 work on singularities of nonlinear elliptic systems, which relies on the reverse Hölder property of that system, was dedicated to N. G. Meyers on the occasion of his 80th birthday.⁴

References

Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 21, 2026