Math 2.0

Partial differential equations Elliptic equations Theorem

Krylov–Safonov estimate

Builds on Elliptic partial differential equationSobolev spaceHölder continuityHarnack's inequality

Statement

Let Ω⊆ℝn\Omega \subseteq \R^n be open and let 0<λ≤Λ0 < \lambda \le \Lambda. Let aij=aji ⁣:Ω→ℝa_{ij} = a_{ji} \colon \Omega \to \R be measurable functions with λ∣ξ∣2≤∑i,j=1naij(x)ξiξj≤Λ∣ξ∣2for all x∈Ω and ξ∈ℝn.\lambda \abs{\xi}^2 \le \sum_{i,j=1}^n a_{ij}(x) \xi_i \xi_j \le \Lambda \abs{\xi}^2 \quad \text{for all } x \in \Omega \text{ and } \xi \in \R^n. Let u∈Wloc2,n(Ω)u \in W^{2,n}_{\mathrm{loc}}(\Omega), a local Sobolev space, satisfy ∑i,jaij∂i∂ju=0\sum_{i,j} a_{ij} \partial_i \partial_j u = 0 almost everywhere in Ω\Omega. There are constants C>0C > 0 and α∈(0,1)\alpha \in (0,1), depending only on nn, λ\lambda and Λ\Lambda, such that for every ball B2r(x0)B_{2r}(x_0) with closure in Ω\Omega:

  1. (Harnack inequality) If u≥0u \ge 0 in B2r(x0)B_{2r}(x_0), then sup⁡Br(x0)u≤Cinf⁡Br(x0)u\sup_{B_r(x_0)} u \le C \inf_{B_r(x_0)} u.
  2. (Hölder estimate) ∣u(x)−u(y)∣≤C(∣x−y∣/r)αsup⁡B2r(x0)∣u∣\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∈Br(x0)x, y \in B_r(x_0).

Remarks

The equation is an elliptic equation in non-divergence form and the coefficients are only measurable, so the energy methods of De Giorgi, Nash and Moser do not apply. The proof uses the Aleksandrov–Bakelman–Pucci maximum principle to show that a supersolution which is large on a fixed fraction of a ball is bounded below on a smaller ball. A covering argument with dyadic cubes in the style of Calderón and Zygmund then turns this into a decay of the measure of level sets, which gives the Harnack inequality. The Hölder continuity estimate follows from the Harnack inequality applied to sup⁡u−u\sup u - u and u−inf⁡uu - \inf u on shrinking balls. For aij=δija_{ij} = \delta_{ij} the first part is the classical Harnack inequality for harmonic functions.

Going deeper

For a student

This is the non-divergence analogue of the De Giorgi–Nash–Moser theorem. It is the starting point of the regularity theory of fully nonlinear elliptic equations F(D2u)=0F(D^2 u) = 0: differences of solutions satisfy linear equations with measurable coefficients, so they are Hölder continuous. With concavity of FF this leads to the Evans–Krylov theorem on C2,αC^{2,\alpha} regularity. Caffarelli extended the theory to viscosity solutions.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Elliptic partial differential equation, concept, no evidence levelElliptic partial diff…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHölder continuity, concept, no evidence levelHölder continuityconceptSobolev space, concept, no evidence levelSobolev spaceconceptUnit disc, object, no evidence levelUnit discobjectTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptPoisson kernel, definition, no evidence levelPoisson kerneldefinitionHarnack's inequality, statement, no evidence levelHarnack's inequalitystatementKrylov–Safonov estimate, statement, no evidence levelKrylov–Safonov estima…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Elliptic partial differential equation concept
  2. Euclidean space object
  3. Function concept
  4. Hölder continuity concept
  5. Sobolev space concept
  6. Unit disc object
  7. Total derivative concept
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Differentiable function of several variables concept
  11. Partial derivative concept
  12. Vector field concept
  13. Gradient concept
  14. Divergence concept
  15. Laplacian concept
  16. Harmonic function concept
  17. Poisson kernel definition
  18. Harnack's inequality statement
  19. Krylov–Safonov estimate statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Harnack%27s_inequality
About this page
Tag5YC2Y65NZX
URIm2://5YC2Y65NZX
Kindstatement, theorem
WritingW0 stub
MSC35J15, 35B65, 35B45
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AboutElliptic partial differential equation, Hölder continuity
UsesElliptic partial differential equation, Sobolev space, Hölder continuity, Harnack's inequality
Changed2 November 2026

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