A relation between the Dirichlet and the Regularity problem for Parabolic equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910959335899136 |
|---|---|
| author | Dindoš, Martin Nyström, Erika |
| author_facet | Dindoš, Martin Nyström, Erika |
| contents | We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ Ω= \mathcal O \times \mathbb R $, where the base $ \mathcal O \subset \mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables.
In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*)_{p'} $).
We show that this holds if for at least of $q\in(1,\infty)$ the problem $ (R_L)_{q} $ is solvable. That is, we establish a duality/dichotomy result: Dirichlet solvability implies Regularity solvability in the dual $L^p$ range, or the Regularity problem is not solvable in any $L^p$.
Results like these were only known in the elliptic settings (Kenig-Pipher (1993) and Shen (2006)) but are new for parabolic PDEs. Our result is one of the key components needed for the recent advancement of Dindoš, Li and Pipher in understanding solvability of the Regularity problem for operators whose coefficients satisfy certain natural Carleson condition (called also DKP-condition in the elliptic case). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A relation between the Dirichlet and the Regularity problem for Parabolic equations Dindoš, Martin Nyström, Erika Analysis of PDEs Classical Analysis and ODEs 35K10, 35K20 We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ Ω= \mathcal O \times \mathbb R $, where the base $ \mathcal O \subset \mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L)_{p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*)_{p'} $). We show that this holds if for at least of $q\in(1,\infty)$ the problem $ (R_L)_{q} $ is solvable. That is, we establish a duality/dichotomy result: Dirichlet solvability implies Regularity solvability in the dual $L^p$ range, or the Regularity problem is not solvable in any $L^p$. Results like these were only known in the elliptic settings (Kenig-Pipher (1993) and Shen (2006)) but are new for parabolic PDEs. Our result is one of the key components needed for the recent advancement of Dindoš, Li and Pipher in understanding solvability of the Regularity problem for operators whose coefficients satisfy certain natural Carleson condition (called also DKP-condition in the elliptic case). |
| title | A relation between the Dirichlet and the Regularity problem for Parabolic equations |
| topic | Analysis of PDEs Classical Analysis and ODEs 35K10, 35K20 |
| url | https://arxiv.org/abs/2409.09197 |