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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.

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Finite element exterior calculus, homological techniques, and applications

In this paper we provide an answer to a question found in [3], namely when given a quasi-metric p, if one examines all quasi-metrics which are pointwise equivalent to p, does there exist one which is most like an ultrametric (or, equivalently, exhibits an optimal amount of Holder regularity)? The answer, in general, is negative, which we demonstrate by constructing a suitable Rolewicz-Orlicz space

/pdf/optimal-quasi-metrics-in-a-given-pointwise-equivalence-class-2xprtvoiu8.pdf

Optimal Quasi-Metrics in a Given Pointwise Equivalence Class do not Always Exist

/pdf/erratum-to-non-self-adjoint-operators-infinite-determinants-5dnh5dze7h.pdf

Erratum to “Non Self Adjoint Operators, Infinite Determinants, and Some Applications,” Russ. J. Math. Phys. 12, 443–471 (2005)

We give an overview of results concerning certain boundary value problems for elliptic Partial Differential Equations on nonsmooth domains, Clifford algebras, and wavelet techniques, and how these are related. In particular, we consider Laplace’s equation on nonsmooth domains and show how this leads to wavelet analysis of certain Clifford algebra valued singular integral operators. A natural context for stuying these operators is the theory of Hardy spaces of Clifford analytic functions, which we discuss in some detail. General elliptic operators are also considered.

Clifford wavelets, Hardy spaces, and elliptic boundary value problems

We prove new mean value formulas for harmonic functions in polyhedra. Our approach is elementary and utilizes vanishing formulas for holomorphic functions of one complex variable and monogenic functions of one quaternionic or Clifford variable.

Marius Mitrea

Papers

Optimal Quasi-Metrics in a Given Pointwise Equivalence Class do not Always Exist

Erratum to “Non Self Adjoint Operators, Infinite Determinants, and Some Applications,” Russ. J. Math. Phys. 12, 443–471 (2005)

Clifford wavelets, Hardy spaces, and elliptic boundary value problems

New mean-value formulas for harmonic functions in polyhedral domains

Geometric Harmonic Analysis III