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关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Real and Complex Analysis

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Methods Of Modern Mathematical Physics

Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

/pdf/finite-element-exterior-calculus-homological-techniques-and-4129s2mxlu.pdf

Finite element exterior calculus, homological techniques, and applications

Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

https://faculty.missouri.edu/~hofmanns/papers/HLMMY_final.pdf

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

Introduction Singular integrals on Lipschitz submanifolds of codimension one Estimates on fundamental solutions General second-order strongly elliptic systems The Dirichlet problem for the Hodge Laplacian and related operators Natural boundary problems for the Hodge Laplacian in Lipschitz domains Layer potential operators on Lipschitz domains Rellich type estimates for differential forms Fredholm properties of boundary integral operators on regular spaces Weak extensions of boundary derivative operators Localization arguments and the end of the proof of Theorem 6.2 Harmonic fields on Lipschitz domains The proofs of the Theorems 5.1-5.5 The proofs of the auxiliary lemmas Applications to Maxwell's equations on Lipschitz domains Analysis on Lipschitz manifolds The connection between $d_\partial$ and $d_{\partial\Omega}$ Bibliography.

Marius Mitrea

Papers

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains

Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds

Boundary layer methods for Lipschitz domains in Riemannian manifolds

Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem