on Moser iteration

Overview

This post provides a rigorous examination of Moser Iteration, a foundational bootstrap technique in the regularity theory of Partial Differential Equations. Developed by Jürgen Moser (Moser, 1961) to provide an alternative proof of the De Giorgi-Nash theorem, the method utilizes the synergy between Caccioppoli-type energy estimates and Sobolev embeddings. It allows one to “pump” the integrability of a sub-solution from $L^2$ to $L^{\infty }$, ultimately yielding local boundedness and the celebrated Harnack Inequality.

🏷️ The Setting: Rough Coefficients and Divergence Form

The power of the Moser technique lies in its ability to handle second-order elliptic operators with measurable coefficients. We consider:

$$\mathcal{L}u=-\sum _{i,j=1}^n{\partial }_i(a_{ij}(x){\partial }_{j}u)=0$$

where $a_{ij}\in L^{\infty }(\Omega )$ are only assumed to be bounded and uniformly elliptic:

$$\lambda |\xi {|}^2\le \sum _{i,j=1}^n{a}_{ij}(x){\xi }_i{\xi }_j\le \Lambda |\xi {|}^2,\forall \xi \in {\mathbb{R}}^n$$

The Divergence Mandate

The divergence structure is non-negotiable for Moser iteration. It enables the use of integration by parts to convert second-order information into first-order gradient control. For non-divergence form equations ($a_{ij}{\partial }_{ij}u=f$), energy methods fail, and one must instead employ the Krylov-Safonov theory based on the Alexandroff-Bakelman-Pucci (ABP) estimate.

🏷️ The Analytical Engine: The Energy Bootstrap

The iteration proceeds by transforming a local $L^p$ bound into a local $L^{p\chi }$ bound for some $\chi >1$. This is achieved by testing the equation with a power of the solution, $u^{\beta }$, and a cutoff function $\eta$.

The Caccioppoli Inequality

For a non-negative sub-solution $u$, testing with $v={\eta }^2{u}^{\beta }$ yields the energy control:

$${\int }_{\Omega }{\eta }^2|\nabla u^{(\beta +1)/2}{|}^{2}dx\le C{\int }_{\Omega }{u}^{\beta +1}|\nabla \eta {|}^{2}dx$$

This inequality shows that the “energy” of a power of $u$ is controlled by the $L^2$ norm of that same power on a slightly larger domain.

🏷️ Main Theorem: Local Boundedness

Local Sup-Norm Estimate

Let $u\in H^1(\Omega )$ be a non-negative sub-solution of $\mathcal{L}u\le 0$ in $\Omega \subset {\mathbb{R}}^n$. For any ball $B_{2R}⋐\Omega$ and any $p>0$, there exists a constant $C(n,\lambda ,\Lambda ,p)$ such that:

$$\Vert u{\Vert }_{L^{\infty }(B_R)}\le \frac{C}{R^{n/p}}\Vert u{\Vert }_{L^p(B_{2R})}$$

Proof: The Iterative Limit ( $n>2$)

To prove the $L^{\infty }$ bound, we construct a sequence of nested balls $B_{R_k}$ with $R_k=R+2^{-k}R$ and integrability exponents $p_k=p{\chi }^k$ (where $\chi =\frac{n}{n-2}$ is the Sobolev gain).

1. The Single Step: Applying the Sobolev inequality $\Vert v{\Vert }_{2\chi }\le C\Vert \nabla v{\Vert }_2$ to the Caccioppoli inequality on the $k$-th ball:

$$\Vert u{\Vert }_{p_{k+1},B_{R_{k+1}}}\le (C\cdot 4^k{\chi }^k{)}^{1/p_k}\Vert u{\Vert }_{p_k,B_{R_k}}$$

The factor $4^k$ arises from the gradient of the cutoff function ${\eta }_k$, which must scale as $(R_k-R_{k+1}{)}^{-1}\approx 2^k/R$.

2. The Iterative Product: Iterating this relation $k$ times:

$$\Vert u{\Vert }_{p_{k+1},B_{R_{k+1}}}\le \left(\prod _{j=0}^k(C\cdot 4^j{\chi }^j{)}^{1/p_j}\right)\Vert u{\Vert }_{p,B_{2R}}$$

3. Convergence of the Constant: Taking the logarithm of the product transforms it into a series:

$$\log (\text{Const})=\sum _{j=0}^{\infty }\frac{1}{p{\chi }^j}\log (C\cdot 4^j{\chi }^j)$$

Since $p_j=p{\chi }^j$ grows geometrically, this series converges. This implies the product of constants remains finite as $k\to \infty$.

4. Conclusion: In the limit, the $L^{p_k}$ norm on the shrinking balls converges to the $L^{\infty }$ norm on $B_R$, yielding the desired bound.

🏷️ The Critical Case: Dimension $n=2$

In two dimensions, the Sobolev gain $\chi$ is formally infinite, which would seemingly break the discrete steps of the iteration.

Proof: The BMO and Exponential Route

For $n=2$, we exploit the fact that $H^1\hookrightarrow L^q$ for all $q<\infty$.

1. Variable Gain: We can choose a fixed gain, say $\chi =2$. The Sobolev constant $C_q$ in $\Vert v{\Vert }_q\le C_q\Vert \nabla v{\Vert }_2$ grows as $\sqrt{q}$. Plugging this into the log-series:

$$\sum \frac{1}{p_j}\log (\sqrt{p_j})=\sum \frac{\log (p{2}^j)}{2p{2}^j}<\infty$$

The convergence is preserved, ensuring the $L^{\infty }$ bound holds.

2. The BMO Insight: A more modern approach recognizes that the energy estimate implies $\nabla \log u\in L^2$. In 2D, this places $\log u$ in the space of Bounded Mean Oscillation (BMO). The John-Nirenberg Theorem then guarantees that $u$ is locally integrable to any power, providing the necessary “launchpad” for the iteration.

🏷️ Scope and Constraints

The Harnack Inequality

By applying the iteration to both $u$ and $1/u$, Moser derived the Harnack Inequality: ${\sup }_{B_R}u\le C{\inf }_{B_R}u$. This implies that positive solutions cannot oscillate wildly and are, in fact, Hölder continuous.

Constraints and Failure Cases

The "Non-Divergence" Barrier

The Moser iteration is fundamentally tethered to the divergence structure. This is due to the nature of the $H^1$ energy space:

1. Lack of Integration by Parts: In non-divergence form equations, $\sum a_{ij}{\partial }_{ij}u=0$, there is no natural way to transfer a derivative to a test function. Thus, we cannot obtain the Caccioppoli inequality that powers the bootstrap.
2. Low Regularity of Coefficients: If the coefficients $a_{ij}$ are only $L^{\infty }$, a solution to a non-divergence form equation may not even possess a gradient in $L^2$. This “low integrability” of the second derivatives prevents the use of Sobolev embeddings.

• The Solution (Krylov-Safonov): For rough coefficients in non-divergence form, one must switch from the analytical bootstrap (Moser) to the geometric maximum principle (Krylov-Safonov (Krylov & Safonov, 1980)). This method uses the ABP (Alexandroff-Bakelman-Pucci) Estimate to show that if a solution is large at a point, it must be large on a set of “positive measure” near the boundary of its convex envelope. See the definitive account in Caffarelli & Cabré [@caffarelli1995fully].
• Dimension $n=2$: The Sobolev gain $\chi$ becomes formally infinite. As detailed in the proof above, this critical case is resolved via BMO theory and the John-Nirenberg inequality.
• Smoothness vs. Structure: If the coefficients are $C^{0,1}$ (Lipschitz), any non-divergence operator can be rewritten in divergence form plus a lower-order drift. In that case, Moser iteration is again applicable. The limitation is specifically for rough coefficients in non-divergence form.

📝 Notes

• The “Nested Ball” method is a strategy to handle the trade-off between domain size and integrability gain.
• This technique shares the “growth tracking” logic found in on Grönwall’s inequality, but operates on the scale of $L^p$ exponents rather than time.

🔗 See Also

• on convergence of graph Laplacian to manifold’s Laplacian --- Moser estimates ensure the equicontinuity of discrete eigenfunctions in the limit.
• on Weyl’s asymptotic law --- Eigenvalue growth is governed by the same Sobolev embedding constants that power the Moser bootstrap.

📚 References

🐻  Krylov, N.V. & Safonov, M.V. 1980. A property of the solutions of elliptic equations with measurable coefficients. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 44(1), 161–175.

🐻  Moser, J. 1961. On Harnack’s theorem for elliptic differential equations. Communications on Pure and Applied Mathematics 14(3), 577–591.