the De Giorgi-Nash-Moser theorem
/ deh-JOR-jee nash MO-zer /
The Schauder and Calderon-Zygmund estimates both lean on the coefficients being smooth or at least Holder continuous. But nature is often not so tidy — composite materials, porous rock, and turbulent media have conductivities that jump wildly and are merely bounded and measurable, with no continuity at all. For such operators the gain of two derivatives is simply false. The De Giorgi-Nash-Moser theorem is the deep result that rescues regularity in this rough regime: a weak solution of a uniformly elliptic divergence-form equation with bounded measurable coefficients is automatically Holder continuous, even though the coefficients are not.
Precisely, for - div(A(x) grad u) = 0 with A uniformly elliptic but only bounded and measurable, any weak solution u is locally C^(0,alpha) for some exponent alpha greater than zero that depends only on the dimension and the ellipticity constants lambda and Lambda — not on any modulus of continuity of A (there is none). Three people reached this independently and differently around 1957-1961: Ennio De Giorgi via clever level-set energy inequalities (truncating u at heights and tracking how the energy of the truncated piece shrinks), John Nash via a parabolic-flow and entropy argument, and Jurgen Moser via an iteration that bootstraps higher and higher integrability of u from a reverse-Holder inequality on dyadic balls — Moser iteration — which also yields the Harnack inequality directly. The miracle is that a uniform positive Holder exponent emerges from data with NO regularity, purely from the elliptic structure and the two-sided bound.
This theorem is the foundation stone of modern nonlinear elliptic theory, and it solved Hilbert's nineteenth problem (regularity of minimisers of analytic variational integrals). It is indispensable wherever coefficients cannot be assumed smooth: homogenisation of composites, equations with rough or random media, and the analysis of minimal surfaces and harmonic maps after one freezes the metric. The Harnack inequality it delivers — comparing the maximum and minimum of a positive solution on a ball with a constant depending only on lambda, Lambda, and the dimension — is the qualitative heart of the result, and it implies the Holder continuity at once.
Imagine heat in a checkerboard material where the conductivity is 1 on the black squares and 100 on the white squares, jumping discontinuously across every edge. The steady temperature solves - div(A grad u) = 0 with this wildly discontinuous A. No Schauder estimate applies. Yet De Giorgi-Nash-Moser guarantees the temperature is still Holder continuous, with an exponent depending only on the ratio 100 and the dimension — the solution cannot have a jump even though the material does.
A discontinuous medium, a continuous solution — Holder regularity that depends only on the ellipticity bounds.
The theorem gives Holder continuity of u itself, NOT of its derivatives — with merely measurable A you generally cannot get even one continuous derivative, let alone two. It is also fundamentally a divergence-form, scalar-equation result; the non-divergence analogue (the Krylov-Safonov theorem) is separate, and for SYSTEMS the conclusion can fail outright (De Giorgi's own later counterexamples).