Boundary value problems for 0-elliptic operators
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913602265415680 |
|---|---|
| author | Usula, Marco |
| author_facet | Usula, Marco |
| contents | Let $X$ be a manifold with boundary, and let $L$ be a 0-elliptic operator on X which is semi-Fredholm essentially surjective with infinite-dimensional kernel. Examples include Hodge Laplacians and Dirac operators on conformally compact manifolds. We construct left and right parametrices for L when supplemented with appropriate elliptic boundary conditions. The construction relies on a new calculus of pseudodifferential operators on functions over both $X$ and $\partial X$, which we call the "symbolic 0-calculus". This new calculus supplements the ordinary 0-calculus of Mazzeo--Melrose, enabling it to handle boundary value problems. In the original 0-calculus, operators are characterized as polyhomogeneous right densities on a blow-up of $X^2$. By contrast, operators in the symbolic 0-calculus are characterized (locally near each point of the boundary of the diagonal) as quantizations of polyhomogeneous symbols on appropriate blown-up model spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary value problems for 0-elliptic operators Usula, Marco Analysis of PDEs Differential Geometry 53-00 Let $X$ be a manifold with boundary, and let $L$ be a 0-elliptic operator on X which is semi-Fredholm essentially surjective with infinite-dimensional kernel. Examples include Hodge Laplacians and Dirac operators on conformally compact manifolds. We construct left and right parametrices for L when supplemented with appropriate elliptic boundary conditions. The construction relies on a new calculus of pseudodifferential operators on functions over both $X$ and $\partial X$, which we call the "symbolic 0-calculus". This new calculus supplements the ordinary 0-calculus of Mazzeo--Melrose, enabling it to handle boundary value problems. In the original 0-calculus, operators are characterized as polyhomogeneous right densities on a blow-up of $X^2$. By contrast, operators in the symbolic 0-calculus are characterized (locally near each point of the boundary of the diagonal) as quantizations of polyhomogeneous symbols on appropriate blown-up model spaces. |
| title | Boundary value problems for 0-elliptic operators |
| topic | Analysis of PDEs Differential Geometry 53-00 |
| url | https://arxiv.org/abs/2412.06084 |