Geometric regularity for elliptic equations in double-divergence form

Leitão, Raimundo; Pimentel, Edgard A.; Santos, Makson S.

doi:10.2140/apde.2020.13.1129

Cited by 2 publications

(1 citation statement)

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“…Analogous results have been obtained in [10] and [11] under the assumption that the matrix A satisfies the Dini mean oscillation condition, which is weaker than the classical Dini condition. Note also the paper [23], where some additional regularity of solutions along level sets has been established. In the papers [1], [2] some interesting counter-examples were constructed and the so-called renormalized solutions were studied, in particular, an example was constructed of a positive definite and continuous diffusion matrix A for which the equation ∂ x i ∂ x j (a ij ̺) = 0 has a locally unbounded solution.…”

Section: Introductionmentioning

confidence: 99%

Zvonkin's transform and the regularity of solutions to double divergence form elliptic equations

Богачев¹,

Röckner²,

Шапошников³

2022

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We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on R d . It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to a power p > d. We establish new estimates for the L p -norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.

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