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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

We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor g jk dx j ⊗ desk has low regularity. Under the assumption that |g jk (x) - g jk (y)| ≤ Cω(|x - y|), where the modulus of continuity ω satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with L P boundary data, for sharp ranges of p's and with optimal nontangential maximal function estimates.

/pdf/potential-theory-on-lipschitz-domains-in-riemannian-172rv4os2z.pdf

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

We show that the boundedness of the Hardy–Littlewood maximal operator on a Kothe function space X and on its Kothe dual X' is equivalent to the well-posedness of the X-Dirichlet and X'-Dirichlet problems inRn+ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H1, and the Beurling-Hardy space HAp for p∈(1,∞). Based on the well-posedness of the LpLp-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

https://www.ems-ph.org/journals/show_pdf.php?issn=0213-2230&vol=32&iss=3&rank=6

The Dirichlet problem for elliptic systems with data in Köthe function spaces

In this paper, we prove a $H^1$-coercive estimate for differential
forms of arbitrary degrees in semi-convex domains of the Euclidean space.
The key result is an integral identity involving a boundary term in which
the Weingarten matrix of the boundary intervenes, established
for any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whose
outward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$,
the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.

Coercive energy estimates for differential forms in semi-convex domains

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to domains with corners.

Lipschitz Domains, Domains with Corners, and the Hodge Laplacian

Let Lr(Ω) be the usual scale of Sobolev spaces and let ∆N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does (−∆N )− r 2 +iδ map {f ∈ Lq(Ω); ∫ Ω f = 0} isomorphically onto Lr(Ω)/IR ?

http://faculty.missouri.edu/~mitream/sqrt_last.pdf

Marius Mitrea

Papers

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

The Dirichlet problem for elliptic systems with data in Köthe function spaces

Coercive energy estimates for differential forms in semi-convex domains

Lipschitz Domains, Domains with Corners, and the Hodge Laplacian

Complex Powers of the Neumann Laplacian in Lipschitz Domains