Vector Potential Theory on Nonsmooth Domains in R3 and Applications to Electromagnetic Scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.     

Sharp Hodge decompositions in two and three dimensional Lipschitz domainsDécompositions de Hodge optimales pour les domaines lipschitziens en dimensions deux et trois

We identify the optimal range of coefficients s, p for which differential forms with coefficients in the Sobolev space $\text{L}{}^{\mathrm{p}}{}_{\mathrm{s}}\text{(Ω)}$ admit natural Hodge decompositions in arbitrary two and three dimensional Lipschitz domains $\text{Ω}$. To cite this article: D. Mitrea, M. Mitrea, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 109–112     

Domain sensitivity analysis of the acoustic far‐field pattern

We consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd.     

Problems of buffon type with multiple intersections for lattices of parallelograms

In this paper we study problems of Buffon type with multiple intersections for lattices of parallelograms and as body test a line segment.     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

     

Linear elliptic boundary value problems in varying domains

The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results.     

Polylactide compositions. II. Correlation between morphology and main properties of PLA/calcium sulfate composites

Starting from calcium sulfate (gypsum) as fermentation by‐product of lactic acid production process, high performance composites have been produced by melt‐blending polylactide (PLA, L/D isomer ratio of 96:4) and β‐anhydrite II (AII) filler, that is, calcium sulfate hemihydrate previously dehydrated at 500 °C. Characterized by attractive mechanical and thermal properties due to good filler dispersion throughout the polyester matrix, these composites are interesting for potential use as biodegradable rigid packaging. Physical characterization of selected composites filled with 20 and 40 wt % AII has been performed and compared to processed unfilled PLA with similar amorphous structure. State of dispersion of the filler particles and interphase characteristic features have been investigated using light microscopy (LM) and scanning electron microscopy (SEM). Addition of AII did not decrease PLA thermal stability as revealed by thermogravimetry analyses (TGA) and allowed reaching a slight increase of PLA crystallizability during melt crystallization and upon heating from the glassy, amorphous state (DSC). It was found by thermomechanical measurements (DMTA) that the AII filler increased pronouncedly storage modulus (E′) of the composites in comparison with PLA in a broad temperature range. The X‐ray investigations showed stable/unchanged crystallographic structure of AII during processing with molten PLA and in the composite system. The notable thermal and mechanical properties of PLA–AII composites are accounted for by the good filler dispersion throughout the polyester matrix confirmed by morphological studies, system stability, and favorable interactions between components. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2770–2780, 2007