---
tag: 32C3D8FG9V
kind: statement
role: theorem
name: De Giorgi–Nash–Moser theorem
msc: [35J15, 35B65, 35D30]
uses: [SJZX81A783, M4CEZ5K9HP, 012PBA0FMD, B4TTNHSTRX]
about: [B4TTNHSTRX, 012PBA0FMD]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_nineteenth_problem
---

## Statement

Let $\Omega \subseteq \R^n$ be open and let $0 < \lambda \le \Lambda$.
Let $a_{ij} \colon \Omega \to \R$, for $1 \le i, j \le n$, be measurable functions with $\abs{a_{ij}(x)} \le \Lambda$ and
$$\sum_{i,j=1}^n a_{ij}(x) \xi_i \xi_j \ge \lambda \abs{\xi}^2 \quad \text{for all } x \in \Omega \text{ and } \xi \in \R^n.$$
Let $u \in W^{1,2}_{\mathrm{loc}}(\Omega)$ be a [[weak solution|SJZX81A783]] of $\sum_{i,j} \partial_i (a_{ij} \partial_j u) = 0$ in $\Omega$, where $W^{1,2}_{\mathrm{loc}}$ is a local [[Sobolev space|M4CEZ5K9HP]].
There are constants $C > 0$ and $\alpha \in (0, 1)$, depending only on $n$, $\lambda$ and $\Lambda$, such that for every ball $B_{2r}(x_0)$ with closure in $\Omega$:

1. $u$ is essentially bounded on $B_r(x_0)$ and $\displaystyle \sup_{B_r(x_0)} \abs{u} \le C \Big( r^{-n} \int_{B_{2r}(x_0)} u^2 \, dx \Big)^{1/2}$.
2. After change on a null set, $u$ is continuous, and $\abs{u(x) - u(y)} \le C \big(\abs{x - y}/r\big)^{\alpha} \sup_{B_{2r}(x_0)} \abs{u}$ for all $x, y \in B_r(x_0)$.

## Remarks

The point is that the coefficients $a_{ij}$ are only measurable, so no smoothness of the equation is passed on to $u$; the [[Hölder continuity|012PBA0FMD]] comes from ellipticity alone.
De Giorgi and Nash proved the result independently; De Giorgi used truncations at many levels and an iteration on the measure of super-level sets.
Moser later gave a proof by iterating reverse Hölder inequalities for powers of $u$, and he obtained a Harnack inequality on the way.
The theorem solves Hilbert's nineteenth problem.
Let $F$ be smooth and uniformly convex, and let $u$ minimize $\int F(\nabla u)\,dx$ in $W^{1,2}$.
Then the derivatives $\partial_k u$ solve an equation of the form above, so they are Hölder continuous.
The Schauder estimates then make $u$ smooth.
The result fails for systems: De Giorgi gave an example of an elliptic system with bounded measurable coefficients with a discontinuous weak solution.

## Explanations

### L1 human

The theorem looks at the simplest kind of [[elliptic equation|B4TTNHSTRX]], but with very rough coefficients that may jump around.
Even so, every solution is continuous, and in fact Hölder continuous, with bounds that depend only on how elliptic the equation is.
This gives a floor of smoothness that rough data cannot break.

### L2 human

This is the key step of regularity theory for nonlinear elliptic equations in divergence form.
Differentiating a nonlinear equation gives a linear equation for the derivatives whose coefficients are only known to be bounded and elliptic; this theorem then makes the derivatives Hölder continuous.
After that, the Schauder estimates bootstrap the regularity to smoothness.
The parabolic version, also due to Nash and Moser, plays the same role for parabolic equations in divergence form.
The counterpart for equations in non-divergence form is the Krylov–Safonov theorem.