Planar Domain Research Articles

Numerical Identification of Boundary Condition for Reaction-Advection-Diffusion Partial Differential Equation

In this study, we consider a reaction-advection-diffusion partial differential equations (PDEs) in a plane domain with missed boundary data. We applied both the KMF algorithm and the conjugate gradient method to reconstruct the missed data by using the spectral element method. Several numerical examples were given illustrating the convergence of the used algorithms.

A note on the space of all Toeplitz operators

Abstract We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

A reverse Faber-Krahn inequality for the magnetic Laplacian

We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).

On the Hardy number of Koenigs domains

This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.

Logarithmically refined Gagliardo–Nirenberg interpolation and application to blow-up exclusion in a singular chemotaxis–consumption system

A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in R 2 {\mathbb {R}}^2 . It is based on an isoparametric C 0 C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

First mixed Laplace eigenfunctions with no hot spots

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in R n \mathbb {R}^n attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.

Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve.

Nanoscale View of Alignment and Domain Growth in a Hexagonal Columnar Liquid Crystal.

Highly ordered liquid crystalline (LC) phases have important potential for organic electronics. We studied the molecular alignment and domain structure in a columnar LC thin film with nanometer resolution during in situ heating using four-dimensional scanning transmission electron microscopy (4D STEM). The initial disordered vapor-deposited LC glass thin film rapidly ordered at its glass transition temperature into a hexagonal columnar phase with small (<10 nm), well-aligned, planar domains (columns oriented parallel to the surface). Upon further heating, the domains coarsen via bulk diffusion, then the film crystallizes, then finally transforms back to an LC phase at an even higher temperature. The LC phase at high temperature shows straight columns of molecules, which we attribute to structure inherited from the intermediate crystalline phase. Nanoscale 4D STEM offers direct insight into the mechanisms of domain reorganization, and intermediate crystallization is a potential approach to manipulate orientational order and texture at the nano- to mesoscale in LC thin films.

Birkhoff attractors of dissipative billiards

Abstract We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

A note on kernel functions of Dirichlet spaces

For a planar domain Ω, we consider the Dirichlet spaces with respect to a base point ζ∈Ω and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on Ω×Ω. This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.

Investigation on spectroscopy characteristics of different metamorphic degrees of coal-based graphite

To gain insights into the spectroscopy characteristics from coal to graphite, we investigated different metamorphic degrees of coal-based graphite which were collected from Hunan Province China. In this paper, by means of X-ray diffraction (XRD), scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), and Raman spectroscopy, graphite with different metamorphism degrees has been characterized to explore the evolution of macromolecular structure of organic matter during graphitization. The results show that with the increase of metamorphism degree, the 002-diffraction peak of the series of samples gradually shrinks and narrows, and the peak intensity becomes stronger, indicating that the microcrystalline structure gradually becomes regular and ordered. As the degree of graphitization increase, the uniformity of particle size in coal samples observed gradually increases, and the morphology becomes more regular, transitioning from disordered and irregular shapes to a structured large-scale flake pattern. The crystallinity improves, and the massive coal particles gradually coalesce into large plate crystals, with the inter-particle pores gradually closing. The graphite structure becomes increasingly evident. The FTIR spectra show that as the degree of graphitization increases, the peak at 1,581 cm−1 corresponding to C=C vibrations gradually intensifies. Some inert functional groups are retained throughout the graphitization process. The pores between coal particles gradually close, and the morphology of graphite particles becomes more regular and ordered. Additionally, during the graphitization process, structures similar to carbon nanotubes may develop. Throughout the structural transformation from coal macromolecules to graphite crystals, the size of the sp2 planar domains in single-layer graphene increases, and the lattice structure of carbon atoms gradually enlarges. These findings contribute to a comprehensive understanding of the properties and characteristics of coal-derived graphite, and can provide theoretical reference and basis for the metallogenic mechanism of coal-derived graphite and the efficient utilization of coal.

Colloidal 2D Layered SiC Quantum Dots from a Liquid Precursor: Surface Passivation, Bright Photoluminescence, and Planar Self-Assembly.

We report the bottom-up synthesis of colloidal two-dimensional (2D) layered silicon carbide (SiC) quantum dots with a cubic structure, lateral size of 5-10 nm, ⟨110⟩ exfoliation to few atomic layers, and surface passivation with 1-dodecene. Samples shielded from oxygen and plasma-annealed for purity exhibit narrow blue photoluminescence (PL) with quantum yields (QYs) over 60% in exceptional cases, while unshielded nanocrystals (NCs) exhibit broad blue/green/white PL with 10-15% QY. The latter scenario is attributed to excess surface carbon and oxygen accrued during synthesis and processing, with size separation through ultracentrifugation revealing size-dependent impurity emission. In contrast, the shape of the bright narrow blue PL shows little variation with NC size, while in both scenarios, the maximum QY occurs near four atomic layers. When dried under heat, the disk-like NC suspensions are observed to aggregate into microscale domains, with further self-assembly into planar superlattice domains with common crystalline orientation. The results are compared with photophysical simulations and bring clarity to the broad emission commonly reported for top-down approaches, while inspiring bottom-up schemes directed at improved material quality.

Mappings of finite distortion on metric surfaces

AbstractWe investigate basic properties of mappings of finite distortion$$f:X \rightarrow \mathbb {R}^2$$ f : X → R 2 , where X is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant $$f:X \rightarrow \mathbb {R}^2$$ f : X → R 2 with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if f is moreover injective then $$f^{-1}$$ f - 1 is a Sobolev map.

NeRF-Texture: Synthesizing Neural Radiance Field Textures.

Texture synthesis is a fundamental problem in computer graphics that would benefit various applications. Existing methods are effective in handling 2D image textures. In contrast, many real-world textures contain meso-structure in the 3D geometry space, such as grass, leaves, and fabrics, which cannot be effectively modeled using only 2D image textures. We propose a novel texture synthesis method with Neural Radiance Fields (NeRF) to capture and synthesize textures from given multi-view images. In the proposed NeRF texture representation, a scene with fine geometric details is disentangled into the meso-structure textures and the underlying base shape. This allows textures with meso-structure to be effectively learned as latent features situated on the base shape, which are fed into a NeRF decoder trained simultaneously to represent the rich view-dependent appearance. Using this implicit representation, we can synthesize NeRF-based textures through patch matching of latent features. However, inconsistencies between the metrics of the reconstructed content space and the latent feature space may compromise the synthesis quality. To enhance matching performance, we further regularize the distribution of latent features by incorporating a clustering constraint. In addition to generating NeRF textures over a planar domain, our method can also synthesize NeRF textures over curved surfaces, which are practically useful. Experimental results and evaluations demonstrate the effectiveness of our approach.

On the critical points of solutions of PDE in non-convex settings: The case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely{−Δu=λeu in Ωu=0 on ∂Ω, where Ω⊂R2 is a bounded smooth domain and λ>0 is a small parameter.

Weighted estimates for the Bergman projection on planar domains

We investigate weighted Lebesgue space estimates for the Bergman projection on a simply connected planar domain via the domain’s Riemann map. We extend the bounds which follow from a standard change-of-variable argument in two ways. First, we provide a regularity condition on the Riemann map, which turns out to be necessary in the case of uniform domains, in order to obtain the full range of weighted estimates for the Bergman projection for weights in a Békollè-Bonami-type class. Second, by slightly strengthening our condition on the Riemann map, we obtain the weighted weak-type (1,1) estimate as well. Our proofs draw on techniques from both conformal mapping and dyadic harmonic analysis.

Spectral structure of the Neumann–Poincaré operator on axially symmetric functions

We consider the Neumann–Poincaré operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when it is restricted to axially symmetric functions. If the boundary of the domain is smooth, we show that there are infinitely many axially symmetric eigenfunctions and derive Weyl-type asymptotics of the corresponding eigenvalues. We also derive the leading order terms of the asymptotic limits of positive and negative eigenvalues. The coefficients of the leading order terms are related to the convexity and concavity of the domain. If the boundary of the domain is less regular, we derive decay estimates of the eigenvalues. The decay rate depends on the regularity of the boundary. We also consider the domains with corners and prove that the essential spectrum of the Neumann–Poincaré operator on the axially symmetric three-dimensional domain is non-trivial and contains that of the planar domain.