The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces on a bounded Lipschitz domain in R 3 , with p in a neighborhood of 2. This system is obtained by adding the semilinear term |u|u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well-posedness results for the Dirichlet and Neumann problems in L p -based Besov spaces on bounded Lipschitz domains in R n (n ≥ 3) are also presented. Then, using integral potential operators, we show the well-posedness in L 2 -based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in R n (n ≥ 3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some L p -based Sobolev spaces. Next, we use the well-posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in L p -based Besov spaces, with p ∈ (2 − , 2 + ) and some parameter > 0. KEYWORDSSemilinear Darcy-Forchheimer-Brinkman system, mixed Dirichlet-Neumann problem, L p -based Besov spaces, layer potential operators, Neumann-to-Dirichlet operator, existence and uniqueness INTRODUCTIONBoundary integral methods are a powerful tool to investigate linear elliptic boundary value problems that appear in various areas of science and engineering (see, eg, previous studies 1-5 ). Among many valuable contributions in the field, we mention the well-posedness result of the Dirichlet problem for the Stokes system in Lipschitz domains in R n (n ≥ 3) with boundary data in L 2 -based Sobolev spaces, which have been obtained by Fabes, Kenig, and Verchota 6 by using a layer potential analysis. Also, Mitrea and Wright 7 obtained the well-posedness results for Dirichlet, 7780Neumann, and transmission problems for the Stokes system on arbitrary Lipschitz domains in R n (n ≥ 2), with data in Sobolev and Besov-Triebel-Lizorkin spaces. By using a boundary integral method, Mitrea and Taylor 5 obtained well-posedness results for the Dirichlet problem for the Stokes system on arbitrary Lipschitz domains on a compact Riemannian manifold, with boundary data in L 2 . Their results extended the results of Fabes, Kenig, and Verchota 6 from the Euclidean setting to the case of compact Riemannian manifolds. Continuing the study of Mitrea and Taylor 5 , Dindos and Mitrea 3 developed a layer potential analysis to obtain existence and uniqueness results for the Poiss...
Microarray image processing leads to the characterization of gene expression levels simultaneously, for all cellular transcripts (mRNAs) in a single experiment. The calculation of expression levels for each microarray spot/gene is a crucial step to extract valuable information. By measuring the mRNA levels for the whole genome, the microarray experiments are capable to study functionality, pathological phenotype, and response of cells to a pharmaceutical treatment. The processing of the extensive number of non-homogeneous data contained in microarray images is still a challenge. We propose a density based spatial clustering procedure driven by a level-set approach for microarray spot segmentation together with a complete set of quality measures used to evaluate the proposed method compared with existing approaches for gene expression levels estimation. The set of quality measures used for evaluation include: regression ratios, intensity ratios, mean absolute error, coefficient of variation and fold change factor. We applied the proposed image processing pipeline to a set of microarray images and compared our results with the ones delivered by Genepix, using the aforementioned quality measures. The advantage of our proposed method is highlighted by a selection of up-regulated genes that had been identified exclusively by our approach. These genes prove to add valuable information regarding the biological mechanism activated as a response of Arabidopsis T to pathogen infection.
Metamaterial slabs made of two layers of metallic nano-resonators with three-fold rotational symmetry, arranged in hexagonal lattices separated by a dielectric layer, are theoretically analyzed as chiral metamaterials. The conjugated arrangement of the paired meta-atoms from the first and second layer provides no center of inversion and no mirror-symmetry plane for the analyzed structure. The symmetry properties of the meta-surface lead to a lack of cross conversion between left and right circularly polarized modes through transmission, ensuring a pure chiral response for the metamaterial slab. A negative refractive index due to chirality and pronounced pure optical activity is obtained in the near-infrared range of the spectrum.
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