Report
Some Bourgain-Brézis type solutions via complex interpolation
| العنوان: | Some Bourgain-Brézis type solutions via complex interpolation |
|---|---|
| المؤلفون: | Curcă, Eduard |
| المساهمون: | Faculty of Computer Science (University of Iasi - Al.I.Cuza) (UAIC Iasi) |
| المصدر: | https://hal.science/hal-04097900 ; 2023. |
| بيانات النشر: | HAL CCSD |
| سنة النشر: | 2023 |
| المجموعة: | Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
| مصطلحات موضوعية: | Bourgain-Brézis solutions Fourier multipliers Divergence equation. MSC 2020 classification: 42B15 42B35 46B70, Bourgain-Brézis solutions, Fourier multipliers, Divergence equation. MSC 2020 classification: 42B15, 42B35, 46B70, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] |
| الوصف: | In 2002 Bourgain and Brezis proved that givena vector field $v\in \mathcal{S}^{\prime }(\mathbb{R}^{d})\cap \dot{W}^{1,d}(\mathbb{R}^{d})$ there exists a vector field $u\in L^{\infty }(\mathbb{R}^{d})\cap \dot{W}^{1,d}(\mathbb{R}^{d})$ such that $\operatorname{div}u=\operatorname{div}v$. We prove several results of a similar nature in which we take intoconsideration the Fourier support of the solutions. For instance, in thecase $d\geq 3$ we prove the following: for any vector field $v\in \mathcal{S}^{\prime }(\mathbb{R}^{d})\cap \dot{B}_{q}^{d/p,p}(\mathbb{R}^{d})$ (where $p\in \lbrack 2,\infty )$ and $q\in (1,2) $), with $supp$ $\widehat{v}\subseteq \mathbb{R}^{d}\backslash (-\infty ,0)^{d}$, there exists a vectorfield $u\in L^{\infty }(\mathbb{R}^{d})\cap \dot{B}_{2}^{d/p,p}(\mathbb{R}^{d})$, with $supp$ $\widehat{v}\subseteq \mathbb{R}^{d}\backslash (-\infty,0)^{d}$, such that \begin{equation*}\operatorname{div}u=\operatorname{div}v,\end{equation*}and \begin{equation*}\left\Vert u\right\Vert _{L^{\infty }\cap \dot{B}_{2}^{d/p,p}}\lesssim\left\Vert v\right\Vert _{\dot{B}_{q}^{d/p,p}}.\end{equation*}Our arguments rely on a version of the complex interpolation method combinedwith some ideas of Bourgain and Brezis. |
| نوع الوثيقة: | report |
| اللغة: | English |
| Relation: | hal-04097900; https://hal.science/hal-04097900; https://hal.science/hal-04097900v2/document; https://hal.science/hal-04097900v2/file/interpolation-2023.pdf |
| الاتاحة: | https://hal.science/hal-04097900 https://hal.science/hal-04097900v2/document https://hal.science/hal-04097900v2/file/interpolation-2023.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.6C44147D |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |