On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains

We study integral operators related to a regularized version of the classical Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\infty$ functions.Comment: 23 page

On the Spectrum of Volume Integral Operators in Acoustic Scattering

Volume integral equations have been used as a theoretical tool in scattering theory for a long time. A classical application is an existence proof for the scattering problem based on the theory of Fredholm integral equations. This approach is described for acoustic and electromagnetic scattering in the books by Colton and Kress [CoKr83, CoKr98] where volume integral equations appear under the name "Lippmann-Schwinger equations". In electromagnetic scattering by penetrable objects, the volume integral equation (VIE) method has also been used for numerical computations. In particular the class of discretization methods known as "discrete dipole approximation" [PuPe73, DrFl94] has become a standard tool in computational optics applied to atmospheric sciences, astrophysics and recently to nano-science under the keyword "optical tweezers", see the survey article [YuHo07] and the literature quoted there. In sharp contrast to the abundance of articles by physicists describing and analyzing applications of the VIE method, the mathematical literature on the subject consists only of a few articles. An early spectral analysis of a VIE for magnetic problems was given in [FrPa84], and more recently [Ki07, KiLe09] have found sufficient conditions for well-posedness of the VIE in electromagnetic and acoustic scattering with variable coefficients. In [CoDK10, CoDS12], we investigated the essential spectrum of the VIE in electromagnetic scattering under general conditions on the complex-valued coefficients, finding necessary and sufficient conditions for well-posedness in the sense of Fredholm in the physically relevant energy spaces. A detailed presentation of these results can be found in the thesis [Sa14]. Publications based on the thesis are in preparation. Curiously, whereas the study of VIE in electromagnetic scattering has thus been completed as far as questions of Fredholm properties are concerned, the simpler case of acoustic scattering does not seem to have been covered in the same depth. It is the purpose of the present paper to close this gap

Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains

Suppose that Ω is the open region in ℝn above a Lipschitz graph and let d denote the exterior derivative on ℝn. We construct a convolution operator T which preserves support in Ω is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that dT is the identity on spaces of exact forms with support in Ω. Thus if f is exact and supported in Ω then there is a potential u, given by u = T f, of optimal regularity and supported in Ω, such that du = f. This has implications for the regularity in homogeneous function spaces of the de Rham complex on Ω with or without boundary conditions. The operator T is used to obtain an atomic characterisation of Hardy spaces Hp of exact forms with support in Ω when n/(n + 1) < p ≤ 1. This is done via an atomic decomposition of functions in the tent spaces Tp(ℝn _ ℝ+) with support in a tent T(Ω) as a sum of atoms with support away from the boundary of Ω . This new decomposition of tent spaces is useful, even for scalar valued functions

Analysis of segregated boundary-domain integral equations for mixed variable-coefficient BVPs in exterior domains

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2011 Birkhäuser Boston.Some direct segregated systems of boundary–domain integral equations (LBDIEs) associated with the mixed boundary value problems for scalar PDEs with variable coefficients in exterior domains are formulated and analyzed in the paper. The LBDIE equivalence to the original boundary value problems and the invertibility of the corresponding boundary–domain integral operators are proved in weighted Sobolev spaces suitable for exterior domains. This extends the results obtained by the authors for interior domains in non-weighted Sobolev spaces.The work was supported by the grant EP/H020497/1 ”Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients” of the EPSRC, UK

Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure

Analysis of some localized boundary-domain integral equations for transmission problems with variable coefficients

This is the post-print version of the Article. The official published version can be found at the links below - Copyright @ 2011 Birkhäuser Boston.Some segregated systems of direct localized boundary-domain integral equations (LBDIEs) associated with several transmission problems for scalar PDEs with variable coefficients are formulated and analyzed for a bounded domain composed of two subdomains with a coefficient jump over the interface. The main results established in the paper are the LBDIE equivalence to the original transmission problems and the invertibility of the corresponding localized boundary-domain integral operators in corresponding Sobolev spaces function spaces.This research was supported by the EPSRC grant EP/H020497/1: ”Mathematical analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” and partly by the Georgian Technical University grant in the case of the third author

Mediat. Inflamm.

There is increasing evidence that proteasomes have a biological role in the extracellular alveolar space, but inflammation could change their composition. We tested whether immunoproteasome protein-containing subpopulations are present in the alveolar space of patients with lung inflammation evoking the acute respiratory distress syndrome (ARDS). Bronchoalveolar lavage (BAL) supernatants and cell pellet lysate from ARDS patients (n = 28) and healthy subjects (n = 10) were analyzed for the presence of immunoproteasome proteins (LMP2 and LMP7) and proteasome subtypes by western blot, chromatographic purification, and 2D-dimensional gelelectrophoresis. In all ARDS patients but not in healthy subjects LMP7 and LMP2 were observed in BAL supernatants. Proteasomes purified from pooled ARDS BAL supernatant showed an altered enzyme activity ratio. Chromatography revealed a distinct pattern with 7 proteasome subtype peaks in BAL supernatant of ARDS patients that differed from healthy subjects. Total proteasome concentration in BAL supernatant was increased in ARDS (971 ng/mL perpendicular to 1116 versus 59 perpendicular to 25; P < 0.001), and all fluorogenic substrates were hydrolyzed, albeit to a lesser extent, with inhibition by epoxomicin (P = 0.0001). Thus, we identified for the first time immunoproteasome proteins and a distinct proteasomal subtype pattern in the alveolar space of ARDS patients, presumably in response to inflammation

Convergence analysis of a multigrid algorithm for the acoustic single layer equation

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in \cite{BLP94}. A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator