Planar Domain Research Articles

On the number of critical points of the second eigenfunction of the Laplacian in convex planar domains

In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has large eccentricity then the eigenfunction has exactly two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([13]) and Grieser-Jerison ([10]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [11].

Remarks on the Solution Map for Yudovich Solutions of the Euler Equations

Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $\mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $p\in [1, \infty)$ and is weakly-$*$ continuous in $L^\infty$.

An Estimate for the Steklov Zeta Function of a Planar Domain Derived from a First Variation Formula

We consider the Steklov zeta function \(\zeta _\varOmega \) of a smooth bounded simply connected planar domain \(\varOmega \subset {\mathbb {R}}^2\) of perimeter \(2\pi \). We provide a first variation formula for \(\zeta _\varOmega \) under a smooth deformation of the domain. On the base of the formula, we prove that, for every \(s\in (-1,0)\cup (0,1)\), the difference \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\) is non-negative and is equal to zero if and only if \(\varOmega \) is a round disk (\(\zeta _\mathrm{R}\) is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality \(\zeta _\varOmega (s)-2\zeta _\mathrm{R}(s)\ge 0\) for \(s\in (-\infty ,-1]\cup (1,\infty )\); the latter fact was proved in our previous paper (2018) in a different way. We also provide an alternative proof of the equality \(\zeta '_\varOmega (0)=2\zeta '_\mathrm{R}(0)\) obtained by Edward and Wu (Determinant of the Neumann operator on smooth Jordan curves. Proc Am Math Soc 111(2):357–363, 1991).

InRS: Implementing the indicator function of NURBS-shaped planar domains

We provide an algorithm that implements the indicator function of NURBS-shaped planar domains, tailored to the fast computation on huge point clouds, together with the corresponding Matlab code.

Quantitative microstructural and spectroscopic investigation of inversion domain boundaries in sintered zinc oxide ceramics doped with iron oxide

Abstract It is known that sintering of powders of zinc oxide (ZnO) with small additions of iron oxide results in a ceramic with grains exhibiting a characteristic inversion domain micro-structure with planar inversion domain boundaries (IDBs) on two different habit planes. This study concentrates on a quantitative analysis, by a combination of different transmission electron microscopy methods, of those IDBs that are parallel to {0001} basal planes of the wurtzite structure of ZnO. Electron diffraction and dark-field imaging prove the nature of the inversion. High-resolution annular dark field scanning transmission electron microscopy allows measurement of the rigid body displacements across these IDBs and of the local lattice contraction related to the octahedral interstices that form the boundaries. Energy-selected imaging, electron energy-loss spectroscopy and energy-dispersive X-ray spectroscopy have been combined to determine the chemical composition of the IDBs quantitatively. It is thus shown unambiguously that every such fault consists of precisely one basal plane of octahedral interstices that are completely occupied by Fe3+ ions and that these FeO6 octahedra are themselves contracted along the <0001> direction. A local charge balance model explains the observations.

Perpendicular magnetization anisotropy induced dynamical coherence reduction in stripe domain film

We investigated the magnetization dynamics of the 350 nm permalloy film with in plane domain (IPD), stripe domain (SD), and labyrinth domain (LD) patterns. Experimental and micromagnetic simulation results showed that the change in magnetic domain structure from IPD to LD was due to the increasing perpendicular magnetic anisotropy (PMA) of the film. The magnetization dynamics indicated that the resonant modes of the film strongly depended on the magnetic domain structure. IPD films presented a uniform precession mode. The film with well-regular SD exhibited clear acoustic and optical resonance modes, and the formation of LD suppressed both resonance modes. Finally, the dynamics of magnetization dependent on the domain structure in these films were discussed by using the phenomenological resonance models.

Embedding theorems related to torsional rigidity and principal frequency

We study criteria for the finiteness of the constants in integral inequalities generalizing the Poincaré–Friedrichs inequality and Saint-Venant’s variational definition of torsional rigidity. The Rayleigh–Faber–Krahn isoperimetric inequality and the Saint-Venant–Pólya inequality guarantee the existence of finite constants for domains of finite volume. Criteria for the existence of finite constants for unbounded domains of infinite volume were known only in the cases of planar simply connected and spatial convex domains. We generalize and strengthen some known results and extend them to the case when . Here is one of our results. Suppose that and , where is a compact set and is either a planar domain with uniformly perfect boundary or a spatial domain satisfying the exterior sphere condition. Under these assumptions, a finite constant exists if and only if the integral is finite, where is the distance from the point to the boundary of .

H-Refinement method for toric parameterization of planar multi-sided computational domain in isogeometric analysis

Toric surface patches are a class of multi-sided surface patches that can represent multi-sided domains without mesh degeneration. In this paper, we propose an improved subdivision algorithm for toric surface patches, which subdivides an N-sided toric surface patch into N rational tensor product Bézier surface patches. By the proposed subdivision algorithm, a Ck-continuous spline surface composed of piecewise toric surface patches is subdivided into a set of rational tensor product Bézier surface patches with Gk-continuity. Additionally, after subdivision, toric surface patches are compatible with CAD systems. Combining the subdivision algorithm with the classical knot insertion algorithm of non-uniform rational B-splines, we develop a novel h-refinement scheme for isogeometric analysis with planar toric parameterizations. Several numerical examples are given to demonstrate the effectiveness and numerical stability of the presented method.

On a conjecture of Gustafsson and Lin concerning Laplacian growth

Gustafsson and Lin recently published a significant result concerning Laplacian growth problems that start from a simply connected planar domain. However, the validity of their result depends on the verification of a particular conjecture. This paper provides the missing proof.

Averaging of magnetic fields and applications

In this paper, we estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the nonlinear Ginzburg–Landau energy, and as a byproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian.

Initial solution estimation of one-step inverse isogeometric analysis for sheet metal forming with complex topologies

The one-step inverse isogeometric analysis method has been successfully applied in sheet metal forming with simple geometries. Generally, actual stamping parts frequently contain numerous trimmed NURBS-based CAD surfaces. The key step in sheet metal forming is to unfold the undevelopable CAD model onto a planar domain and obtain a good initial solution. In this study, we estimate the initial solution of stamping parts with complex topologies using an energy-based algorithm. The trimmed surface analysis technique and Nitsche’s method are adopted for trimmed and multi-patch isogeometric analysis. A new coordinate transformation system is used to avoid special treatment for negative angle and vertical wall problems, which are common in metal sheet forming. We demonstrate our algorithm with three examples and compare the results with one-step inverse isogeometric analysis of simple geometry or the traditional finite element method. These examples illustrate the performance of the new method and its applicability for the integration of design and analysis in sheet metal forming with complex topologies.

On the growth of mth derivatives of algebraic polynomials in the weighted Lebesgue space

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.

CONSTRUCTION OF STABILITY DOMAINS FOR LINEAR DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS

The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation $$ \frac{dz}{dt}=c z(t) + a_1 z(t-1) + a_2 z(t-2) + ... + a_n z(t-n), $$ with fixed $c$, $c \in \mathbb{R}$, $a_k \in \mathbb{R}$, $1 \leq k \leq n$, stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains.

The d-bar formalism for the modified Veselov-Novikov equation on the half-plane

We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural \((2+1)\)-dimensional generalization of the \((1+1)\)-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the \(d\)-bar formalism, also known as Pompieu's formula. In addition, we characterize the \(d\)-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.

AN APPLICATION OF THE METHOD OF FUNDAMENTAL SOLUTIONS FOR THE ELASTODYNAMIC PROBLEM

A method of the fundamental solutions (MFS) is employed for the initial boundary value problem for the elastodynamic equation in annular planar domains. Using the Laguerre transformation in time the non-stationary problem is reduced to a sequence of stationary Dirichlet problems with a known sequence of fundamental solutions. The solutions of stationary problems are approximated by a linear combination of the narrowing of the fundamental solutions. By the collocation method from the Dirichlet data, recursive linear systems are obtained for finding the unknown coefficients in the MFS approximations. Presented numerical results confirm the applicability of the proposed method.

Stable anisotropic capillary hypersurfaces in a wedge

<abstract><p>We study a variational problem for hypersurfaces in a wedge in the Euclidean space. Our wedge is bounded by a finitely many hyperplanes passing a common point. The total energy of each hypersurface is the sum of its anisotropic surface energy and the wetting energy of the planar domain bounded by the boundary of the considered hypersurface. An anisotropic surface energy is a generalization of the surface area which was introduced to model the surface tension of a small crystal. We show an existence and uniqueness result of local minimizers of the total energy among hypersurfaces enclosing the same volume. Our result is new even when the special case where the surface energy is the surface area.</p></abstract>

Two Theorems on the Torsional Rigidity of Piezoelectric Beams

In this paper two inequalities are presented for the torsional rigidity of homogeneous monoclinic piezoelectric beams. All results of the paper are based on the Saint-Venant theory of uniform torsion. The cross section of the considered elastic and piezoelectric beams may be simply connected or multiply connected two-dimensional bounded plane domain. Examples illustrate the proven inequality relations.

Constructing diffeomorphisms between simply connected plane domains

Consider a simply connected domain $\Omega\subset\mathbb{R}^2$ with boundary $\partial \Omega$that is given by a smooth function $\varphi:[a,b]\mapsto \mathbb{R}^2$. Our goal is to calculatea diffeomorphism $\Phi:\mathbb{B}_1(0)\mapsto \Omega$, $\mathbb{B}_1(0)$ the open unit disk in $\mathbb{R}^2$.We present two different methods where both methods are able to handle boundaries $\partial \Omega$that are not star-shaped. The first method is based on an optimization algorithm thatoptimizes the curvature of the boundary, and the second method is based on the physical principleof minimizing a potential energy.Both methods construct first a homotopy between the boundary $\partial\mathbb{B}_1(0)$and $\partial \Omega$ and then extend the boundary homotopy to the inside of the domains.Numerical examples show that the method is applicable to a wide variety of domains $\Omega$.

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov—Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.

On singularities of solution of the elasticity system in a bounded domain with angular corner points

This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξμ. We derive the transcendental equations for all 10 possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, that is, tangential and normal direction, respectively, which depends on ξμ. So a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ωcritical are listed such that for interior angles ω < ωcritical the H2‐regularity of solution can be guaranteed. Moreover, the characterization of stress singularities in terms of the inner angle of a corner point is studied, and the regularity results are given.