Partial differential equations Elliptic equations Theorem

De Giorgi–Nash–Moser theorem

Builds on Weak solution Sobolev space Hölder continuity Elliptic partial differential equation

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 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. 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 $\Omega$ :

$u$ is essentially bounded on $B_r(x_0)$ and $\displaystyle \sup_{B_r(x_0)} \abs{u} \le C \Big( r^{-n} \int_{B_{2r}(x_0)} u^2 \, dx \Big)^{1/2}$ .
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 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.

Going deeper

For a student

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.

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Elliptic partial differential equation concept
Hölder continuity concept
Sobolev space concept
Weak solution concept
De Giorgi–Nash–Moser theorem statement

Tag	`32C3D8FG9V`
URI	`m2://32C3D8FG9V`
Kind	statement, theorem
Writing	W0 stub
MSC	35J15, 35B65, 35D30
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About	Elliptic partial differential equation, Hölder continuity
Uses	Weak solution, Sobolev space, Hölder continuity, Elliptic partial differential equation
Changed	2 November 2026