2024/12/08 by Marco Usula, Usula, Marco · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2412.06084
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 ∂ 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 X2. 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.