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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.18272 |
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| _version_ | 1866917885782261760 |
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| author | ter Elst, A. F. M. Ouhabaz, E. M. |
| author_facet | ter Elst, A. F. M. Ouhabaz, E. M. |
| contents | We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(Ω) \] with possibly complex coefficients. We study three problems:
1) Boundedness on $C^ν$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$.
2) Hölder and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$.
3) Poisson bounds for the heat kernel of ${\cal N}$.
We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+κ}$ for some $κ> 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function $G$ of the elliptic operator with Dirichlet boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18272 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators ter Elst, A. F. M. Ouhabaz, E. M. Analysis of PDEs Functional Analysis 35K08, 58G11, 47B47 We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(Ω) \] with possibly complex coefficients. We study three problems: 1) Boundedness on $C^ν$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) Hölder and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$. 3) Poisson bounds for the heat kernel of ${\cal N}$. We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+κ}$ for some $κ> 0$. For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function $G$ of the elliptic operator with Dirichlet boundary conditions. |
| title | Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators |
| topic | Analysis of PDEs Functional Analysis 35K08, 58G11, 47B47 |
| url | https://arxiv.org/abs/2404.18272 |