Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients
We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form ∂α(Aαβij∂βuj)=0. It is known that the Dini continuity of coefficient matrix A=(Aαβij) is essential for the differentiability of solutions. We prove the following results: (a) If A satisfies a condition sli...
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| Format: | Journal article |
| Language: | English |
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Springer
2023
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| _version_ | 1797109581808140288 |
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| author | Nguyen, L |
| author_facet | Nguyen, L |
| author_sort | Nguyen, L |
| collection | OXFORD |
| description | We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form ∂α(Aαβij∂βuj)=0. It is known that the Dini continuity of coefficient matrix A=(Aαβij) is essential for the differentiability of solutions. We prove the following results:
(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,2 of A satisfies
$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $
where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.
(b) If XA,2 = 0, then ∇u ∈ V MO.
(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when ∇u∈L∞∩VMO. |
| first_indexed | 2024-03-07T07:43:40Z |
| format | Journal article |
| id | oxford-uuid:454d0cac-54aa-4e56-965c-97d723abeda9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:43:40Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:454d0cac-54aa-4e56-965c-97d723abeda92023-05-17T07:38:33ZMean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficientsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:454d0cac-54aa-4e56-965c-97d723abeda9EnglishSymplectic ElementsSpringer2023Nguyen, LWe consider gradient estimates for H1 solutions of linear elliptic systems in divergence form ∂α(Aαβij∂βuj)=0. It is known that the Dini continuity of coefficient matrix A=(Aαβij) is essential for the differentiability of solutions. We prove the following results: (a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,2 of A satisfies $ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $ where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO. (b) If XA,2 = 0, then ∇u ∈ V MO. (c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when ∇u∈L∞∩VMO. |
| spellingShingle | Nguyen, L Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title | Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title_full | Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title_fullStr | Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title_full_unstemmed | Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title_short | Mean oscillation gradient estimates for elliptic systems in divergence form with VMO coefficients |
| title_sort | mean oscillation gradient estimates for elliptic systems in divergence form with vmo coefficients |
| work_keys_str_mv | AT nguyenl meanoscillationgradientestimatesforellipticsystemsindivergenceformwithvmocoefficients |