Domain In R2 Research Articles

Boundary estimates of solutions for a class of Monge–Ampère equations

On a smooth strictly convex bounded domain in R2, we establish some boundary estimates of the solution to the complete hyperbolic affine sphere equation. On the unit ball in Rn, we also give some sharp estimates of the solutions of a class of Monge–Ampère equations. As an application, we prove the relative hyperbolic hypersurface constructed by solving two linked Monge–Ampère equations, has constant Gauss–Kronecker curvature and has bounded principal curvatures.

A plane wave discontinuous Galerkin method with a Dirichlet-to-Neumann boundary condition for the scattering problem in acoustics

We consider the numerical solution of an acoustic scattering problem by the Plane Wave Discontinuous Galerkin Method (PWDG) in the exterior of a bounded domain in R2. In order to apply the PWDG method, we introduce an artificial boundary to truncate the domain, and we impose a non-local Dirichlet-to-Neumann (DtN) boundary condition on the artificial curve. To define the method, we introduce new consistent numerical fluxes that incorporate the truncated series of the DtN map. Error estimates with respect to the truncation order of the DtN map, and with respect to mesh width are derived. Numerical results suggest that the accuracy of the PWDG method for the acoustic scattering problem can be significantly improved by using DtN boundary conditions.

Spectrum of a singularly perturbed periodic thin waveguide

We consider a family {Ωε}ε>0 of periodic domains in R2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −ΔΩε on Ωε. The waveguide Ωε is a union of a thin straight strip of the width ε and a family of small protuberances with the so-called “room-and-passage” geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. We prove a (kind of) resolvent convergence of −ΔΩε to a certain operator on the line as ε→0. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of “passages” are appropriately scaled the first spectral gap of −ΔΩε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

Retinal vessel delineation using a brain-inspired wavelet transform and random forest

This paper presents a supervised retinal vessel segmentation by incorporating vessel filtering and wavelet transform features from orientation scores (OSs), and green intensity. Through an anisotropic wavelet-type transform, a 2D image is lifted to a 3D orientation score in the Lie-group domain of positions and orientations R2⋊S1. Elongated structures are disentangled into their corresponding orientation planes and enhanced via multi-orientation vessel filtering. In addition, scale-selective OSs (in the domain of positions, orientations and scales R2⋊S1×R+) are obtained by adding a scale adaptation to the wavelet transform. Features are optimally extracted by taking maximum orientation responses at multiple scales, to represent vessels of changing calibers. Finally, we train a Random Forest classifier for vessel segmentation. Extensive validations show that our method achieves a competitive segmentation, and better vessel preservation with less false detections compared with the state-of-the-art methods.

The Dirichlet problem in a planar domain with two moderately close holes

We investigate a Dirichlet problem for the Laplace equation in a domain of R2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance |ϵ1| one from the other and each one of size |ϵ1ϵ2|. In such a domain, we introduce a Dirichlet problem and we denote by uϵ1,ϵ2 its solution. We show that the dependence of uϵ1,ϵ2 upon (ϵ1,ϵ2) can be described in terms of real analytic maps of the pair (ϵ1,ϵ2) defined in an open neighbourhood of (0,0) and of logarithmic functions of ϵ1 and ϵ2. Then we study the asymptotic behaviour of uϵ1,ϵ2 as ϵ1 and ϵ2 tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ϵ1 and ϵ2.

Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

Abstract We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.

A note on Sobolev isometric immersions below W2,2 regularity

This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W2,2 setting. We show that the Hessian of each coordinate function of a W2,p, p<2, isometric immersion satisfies a low rank property in the almost everywhere sense, in particular, its Gaussian curvature vanishes almost everywhere. Meanwhile, we provide an example of a W2,p, p<2, isometric immersion from a bounded domain of R2 into R3 that has multiple singularities.

Identification and Characterization of the Diverse Stress-Responsive R2R3-RMYB Transcription Factor from Hibiscus sabdariffa L.

Various regulatory proteins play a fundamental role to manage the healthy plant growth under stress conditions. Differential display reverse transcriptase PCR and random amplification of cDNA ends (RACE) was used to explore the osmotic stress-responsive transcripts. We identified and characterized the salt stress-responsive R2R3 type RMYB transcription factor from Hibiscus sabdariffa which has an open reading frame of 690 bp, encoding 229 long chain amino acids. In silico analysis confirmed the conserved R2 and R3 domain as well as an NLS-1 localization site. The deduced amino acids of RMYB shared 83, 81, 80, 79, 72, 71, and 66% homology with Arabidopsis thaliana, Glycine max, Oryza sativa, Zea maize, Malus domestica, Populus tremula × Populus alba, and Medicago sativa specific MYB family, respectively. We observed the gene upregulation in stem, leaf, and root tissue in response to abiotic stress. Furthermore, RMYB gene was cloned into plant expression vector under CaMV35S promoter and transformed to Gossypium hirsutum: a local cotton cultivar. Overexpression of RMYB was observed in transgenic plants under abiotic stresses which further suggests its regulatory role in response to stressful conditions. The RMYB transcription factor-overexpressing in transgenic cotton plants may be used as potential agent for the development of stress tolerant crop cultivars.

Scalar curvature type problem on the three dimensional bounded domain

In this paper we prove an existence result for the nonlinear elliptic problem: Δu = K u5, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝ3 and K is a positive function in Ω¯. Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.

A uniqueness result for a class of non-strictly convex variational problems

Let Ω be a smooth domain in R2, we prove that if g:[0,+∞)→[0,+∞] is convex with g(0)<g(t) whenever t>0 then there exists an unique minimizer u∈C0,1(Ω) of the functional u↦∫Ωg(|∇u|)dxdy among all Lipschitz-continuous functions that assume the same value of u on ∂Ω.

Trans‐scleral delivery of novel anti‐angiogenic small molecule inhibitors of SRPK1

PurposeDevelopment of non‐invasive, novel therapies for wet age‐related macular degeneration (wAMD) and diabetic macular oedema (DME) is hindered by insufficient delivery to the retina. SRPK1 is a novel target that regulates VEGF‐A splicing. We used a rational medicinal chemistry approach to specifically design novel inhibitors with properties required for trans‐scleral delivery.MethodsFreshly enucleated porcine eyes were dissected, and eye tissue was clamped into a scaffold with inhibitors in the upper chamber facing the sclera. Tissue was dissected at 24 h and compound measured by mass spectrometry. Compounds were screened based on structure, potency, selectivity, molecular weight, cLogP and SAR to inform further development. Efficacy and PK were evaluated in vivo in C57/Bl6 mice. New Zealand White rabbits received 3× daily eye drop for 6 days then eyes were dissected, compounds analysed by mass spectrometry.ResultsPotent SRPK1 inhibitors had improved permeability ex vivo compared to pazopanib (SPHINX‐A 1.47 × 10−6 cm/s, Pazopanib 0.07 × 10−6 cm/s p &lt; 0.0005). Modifications of the R1 and R2 domains, based on crystal structure analysis, led to enhanced permeability (SPHINX‐B 3.46 cm/s*10−6). Permeability did not correlate with molecular weight or cLogP but could be affected by additional parameters such as melanin binding. SPHINX‐A was detected at 0.008% of the total applied dose (84 μg) and above its target IC50 value in rabbit eyes. SRPK1 inhibitor eye drops inhibited laser‐CNV (EC50s &lt; 0.5 μM, n = 6–8, p &lt; 0.05, One‐way ANOVA).ConclusionsWith increased specificity, potency and trans‐scleral permeability, novel SRPK1 inhibitors have potential to reach therapeutic levels in posterior eye segments following eye drop administration and improve treatment for patients with wAMD and DME.

Sharp numerical inclusion of the best constant for embedding [formula omitted] on bounded convex domain

In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding H01(Ω)↪Lp(Ω) on a bounded convex domain in R2. We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented.

Large mass boundary condensation patterns in the stationary Keller–Segel system

We consider the boundary value problem{−Δu+u=λeu,inΩ∂νu=0on∂Ω where Ω is a bounded smooth domain in R2, λ>0 and ν is the inner normal derivative at ∂Ω. This problem is equivalent to the stationary Keller–Segel system from chemotaxis.We establish the existence of a solution uλ which exhibits a sharp boundary layer along the entire boundary ∂Ω as λ→0. These solutions have large mass in the sense that ∫Ωλeuλ∼|log⁡λ|.

Optical flow with fractional order regularization: Variational model and solution method

An optical flow variational model is proposed for a sequence of images defined on a domain in R2. We introduce a regularization term given by the L1 norm of a fractional differential operator. To solve the minimization problem we apply the split Bregman method. Extensive experimental results, with performance evaluation, are presented to demonstrate the effectiveness of the new model and method and to show that our algorithm performs favorably in comparison to another existing method. We also discuss the influence of the order α of the fractional operator in the estimation of the optical flow, for 0≤α≤2. We observe that the values of α for which the method performs better depend on the geometry and texture complexity of the image. Some extensions of our algorithm are also discussed.

Arguments towards the construction of a matrix model groundstate

We discuss the existence and uniqueness of wavefunctions for inhomogenoeus boundary value problems associated to x2y2-type matrix model on a bounded domain of ℝ2. Both properties involve a combination of the Cauchy-Kovalewski Theorem and a explicit calculations.

Recovering time-dependent inclusion in heat-conductive bodies using a dynamical probe method

We consider an inverse boundary value problem for the heat equation: ∂tv=divx(γ∇xv) in (0,T)×Ω, where Ω is a bounded domain of R3, and the heat conductivity γ(t,x) admits a surface of discontinuity, which depends on time without any spatial smoothness. The reconstruction and, implicitly, the uniqueness of the moving inclusion based on the knowledge of the Dirichlet-to-Neumann operator is achieved using a dynamical probe method according to the construction of fundamental solutions of the elliptic operator −Δ+τ2⋅, where τ is a large real parameter, and a pair of inequalities relate the data and integrals on the inclusion, in a similar manner to the elliptic case. These solutions depend on the pole of the fundamental solution but also on the large parameter τ, which allows the method to be applied in a very general situation.

Vanishing viscosity limit for the 3D magnetohydrodynamic system with generalized Navier slip boundary conditions

In this paper, we investigate the vanishing viscosity limit problem for the 3D incompressible magnetohydrodynamic (MHD) system in a general bounded smooth domain of R3 with the generalized Navier slip boundary conditions. We also obtain rates of convergence of the solution of viscous MHD to the corresponding ideal MHD. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Mathematical basis for a mixed inverse scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R2 as cross section. To this end, we solve a mixed scattering problem for the Helmholtz equation in R2 where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we set suitable boundary conditions on Γ and ∂D (∂D∈C2). The boundary integral method is employed to study the direct scattering problem. The mathematical basis is given to reconstruct the shape of the crack and the obstacle by using the linear sampling method. The numerical examples are given to show the viability of the method.

Concentrating solutions for an exponential nonlinearity with Robin boundary condition

We study the following Robin boundary value problem (0.1){Δu+ϵ2up−1eup=0,u>0,inΩ;∂u∂n+λu=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, as ϵ→0 and simultaneously λ→+∞, 0<p<2. By using a Lyapunov–Schmidt reduction procedure, we extend for the whole range of exponents 0<p<2, the results of Dávila and Topp (2012), who constructed bubbling solutions to problem (0.1) for p=1.

Blow up solutions for a Liouville equation with Hénon term

We study the following boundary value problem (0.1){Δu+λ|x|2αup−1eup=0,u>0,inΩ;u=0on∂Ω, where Ω is a bounded domain in R2 with smooth boundary, α∈(0,∞)∖N, λ>0 is a small parameter and 0<p<2. We construct the bubbling solution to problem (0.1) which blows up at the origin and at some other m points.