De Giorgi-Nash-Moser theorem

If $u\in W^{1,2}(B_1)$ weakly solves $D_i(a^{ij}{D}_{j}u)=0$ with bounded measurable uniformly elliptic coefficients, then $u$ is locally Hölder continuous. For $0<\theta <1$, some $\alpha \in (0,1)$ and $C$ depending only on the dimension, ellipticity bounds, and $\theta$ satisfy

For a fixed bounded measurable uniformly elliptic operator in divergence form, solutions bounded by one have uniform local Caccioppoli inequalities and Hölder continuity estimates. A diagonal subsequence argument yields a local weak $W^{1,2}$ limit, while the Arzelà-Ascoli theorem gives uniform convergence on a fixed smaller closed Euclidean ball. The weak equation passes to the limit because the coefficients are fixed and bounded. A global supremum normalization does not force the interior limit to be nonzero.

Moser iteration combines a Caccioppoli inequality with the Sobolev embedding theorem to raise an integrability exponent geometrically. On nested balls it turns an $L^p$ bound for a nonnegative subsolution of a uniformly elliptic divergence-form equation into a local supremum bound.

For $q_i=2{\alpha }^i$, $r_i=1/2+2^{-i-1}$, a uniform power gain for elliptic solutions bounds the successive norms by factors $[DR{2}^{i+2}{]}^{{\alpha }^{-i}}$. Their infinite product isIt is finite because both $\sum {\alpha }^{-i}$ and $\sum i{\alpha }^{-i}$ converge. Uniform control of the $L^{q_i}$ norms on $B_{1/2}$ then controls the essential supremum, by comparing with the measure of a set where the function exceeds the proposed bound. A constant independent of $q$ left outside the $2/q$ power at every step would instead give a divergent product.

For $n\ge 3$, a nonnegative entire weak subsolution of a bounded measurable uniformly elliptic divergence-form equation that belongs to $L^p({\mathbb{R}}^n)$ for some $p>1$ is zero almost everywhere. Rescaling the local Moser iteration estimate gives $\Vert u{\Vert }_{L^{\infty }(B_{R/2})}\le C{R}^{-n/p}\Vert u{\Vert }_{L^p(B_R)}$. Let $R\to \infty$.

A Weak Harnack inequality controls a positive integral mean of a nonnegative weak supersolution by its essential infimum. For example, let $n\ge 3$, let $A$ be a bounded measurable uniformly elliptic coefficient matrix, and let $u\in H_{\mathrm{loc}}^1$ be nonnegative with $\mathrm{div}(A\nabla u)\le 0$ in the weak sense. If $B_{4R}⋐\Omega$, then for $0<q<n/(n-2)$,The constant depends on dimension, ellipticity/coefficient bounds and the chosen exponent. The normalization makes the estimate invariant under spatial rescaling. Its infimum is essential because a function in a Sobolev space is initially an almost-everywhere equivalence class; changing a single representative value cannot invalidate the estimate.

This is a one-sided statement for supersolutions, distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions. Combining it with the subsolution bound from Moser iteration yields that latter estimate. If the essential infimum is zero, the left side forces vanishing almost everywhere on the larger ball, giving the mechanism of propagation of zeros by the weak Harnack inequality. Lower-order terms or forcing require the corresponding modified assumptions and error terms.

For a nonnegative weak supersolution with zero forcing, the Weak Harnack inequality makes zero essential infimum on an interior ball imply vanishing almost everywhere on that ball. An overlapping ball then has zero essential infimum too. Chains of overlapping interior balls propagate the vanishing through a connected domain.

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the Weak Harnack inequality implyfor a universal $0<\theta <1$. Iteration gives local Hölder continuity.

If the essential oscillation on concentric balls obeys $\omega (R/2)\le \theta \omega (R)+A{R}^{\alpha }$ with $0<\theta <1$ and $\alpha >0$, dyadic iteration gives $\omega (r)\le C{r}^{\mu }$ for every $0<\mu <\min \{\alpha ,-{\log }_2\theta \}$. The resulting decay gives a Hölder continuous function. Taking the exponent strictly below both decay rates avoids a logarithmic factor when the rates coincide.

A nonnegative weak solution of a uniformly elliptic divergence-form equation satisfieswhenever a fixed larger concentric ball lies in the domain. Combine the subsolution bound from Moser iteration with the Weak Harnack inequality for supersolutions.