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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2301.13657 |
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| _version_ | 1866908376675385344 |
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| author | Geng, Xingri |
| author_facet | Geng, Xingri |
| contents | We discuss the initial boundary value problem for a heat equation in a domain surrounded by a layer. The main features of this problem are twofold: on one hand, the layer is thin compared to the scale of the domain, and on the other hand, the thermal conductivity of the layer is drastically different from that of the bulk; moreover, the bulk is isotropic, but the layer is anisotropic and ``optimally aligned" in the sense that any vector in the layer normal to the interface is an eigenvector of the thermal tensor. We study the effects of the layer by thinking of it as a thickless surface, on which ``effective boundary conditions" (EBCs) are imposed. In the three-dimensional case, we obtain EBCs by investigating the limiting solution of the initial boundary value problem subject to either Dirichlet or Neumann boundary conditions as the thickness of the layer shrinks to zero. These EBCs contain not only the standard boundary conditions but also some nonlocal ones, including the Dirichlet-to-Neumann mapping and the fractional Laplacian. One of the main features of this work is to allow the drastic difference in the thermal conductivity in the normal direction and two tangential directions within the layer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Effective Boundary Conditions for Heat Equation Arising from Anisotropic and Optimally Aligned Coatings in Three Dimensions Geng, Xingri Analysis of PDEs We discuss the initial boundary value problem for a heat equation in a domain surrounded by a layer. The main features of this problem are twofold: on one hand, the layer is thin compared to the scale of the domain, and on the other hand, the thermal conductivity of the layer is drastically different from that of the bulk; moreover, the bulk is isotropic, but the layer is anisotropic and ``optimally aligned" in the sense that any vector in the layer normal to the interface is an eigenvector of the thermal tensor. We study the effects of the layer by thinking of it as a thickless surface, on which ``effective boundary conditions" (EBCs) are imposed. In the three-dimensional case, we obtain EBCs by investigating the limiting solution of the initial boundary value problem subject to either Dirichlet or Neumann boundary conditions as the thickness of the layer shrinks to zero. These EBCs contain not only the standard boundary conditions but also some nonlocal ones, including the Dirichlet-to-Neumann mapping and the fractional Laplacian. One of the main features of this work is to allow the drastic difference in the thermal conductivity in the normal direction and two tangential directions within the layer. |
| title | Effective Boundary Conditions for Heat Equation Arising from Anisotropic and Optimally Aligned Coatings in Three Dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2301.13657 |