Alternating Boundary Conditions Research Articles

Asymptotic Expansions of Solutions to the Poisson Equation with Alternating Boundary Conditions on an Open Arc

Asymptotic Expansions of Solutions to the Poisson Equation with Alternating Boundary Conditions on an Open Arc

Multiple Ising Interfaces in Annulus and 2N-Sided Radial SLE

Abstract We consider critical planar Ising model in annulus with alternating boundary conditions on the outer boundary and free boundary conditions in the inner boundary. As the size of the inner hole goes to zero, the event that all interfaces get close to the inner hole before they meet each other is a rare event. We prove that the law of the collection of the interfaces conditional on this rare event converges in total variation distance to the so-called $2N$-sided radial SLE$_{3}$, introduced by [ 16]. The proof relies crucially on an estimate for multiple chordal SLE. Suppose $(\gamma _{1}, \ldots , \gamma _{N})$ is chordal $N$-SLE$_{\kappa }$ with $\kappa \in (0,4]$ in the unit disc, and we consider the probability that all $N$ curves get close to the origin. We prove that the limit $\lim _{r\to 0+}r^{-A_{2N}}\mathbb{P}[\textrm{dist}(0,\gamma _{j})<r, 1\le j\le N]$ exists, where $A_{2N}$ is the so-called $2N$-arm exponents and $\textrm{dist}$ is Euclidean distance. We call the limit Green’s function for chordal $N$-SLE$_{\kappa }$. This estimate is a generalization of previous conclusions with $N=1$ and $N=2$ proved in [ 25, 26] and [ 32], respectively.

Loop-erased random walk branch of uniform spanning tree in topological polygons

We consider uniform spanning tree (UST) in topological polygons with 2N marked points on the boundary with alternating boundary conditions. In an earlier work by Liu-Peltola-Wu, the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE8. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE2. The conclusion is a generalization of an earlier work by Han-Liu-Wu where the authors derive the scaling limit of the LERW branch of UST when N=2. When N=2, the limiting law is SLE2(−1,−1;−1,−1). However, the limiting law is no longer in the family of SLE2(ρ) process as long as N≥3.

Operator L2-Estimates for Two-Dimensional Problems with Rapidly Alternating Boundary Conditions

Operator L2-Estimates for Two-Dimensional Problems with Rapidly Alternating Boundary Conditions

Homogenization of the optimal control problem for the Dirichlet cost functional and the Poisson state problem with rapidly alternating boundary conditions in critical case

Homogenization of the optimal control problem for the Dirichlet cost functional and the Poisson state problem with rapidly alternating boundary conditions in critical case

Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions.

With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane y>0 with different boundary conditions a and b on the negative and positive x axes. For ab=-+ and f+, they determined the one- and two-point averages of the spin σ and energy ε. Here +,-, and f stand for spin-up, spin-down, and free-spin boundaries, respectively. The case +-+-+⋯, where the boundary condition switches between + and - at arbitrary points, ζ_{1},ζ_{2},⋯ on the x axis was also analyzed. In the first half of this paper a similar study is carried out for the alternating boundary condition +f+f+⋯ and the case -f+ of three different boundary conditions. Exact results for the one- and two-point averages of σ,ε, and the stress tensor T are derived with conformal-invariance methods. From the results for 〈T〉, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is derived for mixed boundary conditions. In the second half of the paper, arbitrary two-dimensional critical systems with mixed boundary conditions are analyzed with boundary-operator expansions. Two distinct types of expansions-away from switching points of the boundary condition and at switching points-are considered. Using the expansions, we express the asymptotic behavior of two-point averages near boundaries in terms of one-point averages. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge. Finally we confirm the consistency of the predictions obtained with conformal-invariance methods and with boundary-operator expansions, in the the first and second halves of the paper.

Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

We consider a spectral homogenization problem for the linear elasticity system posed in a domain $\varOmega $ of the upper half-space $\mathbb{R}^{3+}$ , a part of its boundary $\varSigma $ being in contact with the plane $\{x_{3}=0\}$ . We assume that the surface $\varSigma $ is traction-free out of small regions $T^{\varepsilon }$ , where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function $M(x)$ and a reaction parameter $\beta (\varepsilon )$ that can be very large when $\varepsilon \to 0$ . The size of the regions $T^{\varepsilon }$ is $O(r_{\varepsilon })$ , where $r_{\varepsilon }\ll \varepsilon $ , and they are placed at a distance $\varepsilon $ between them. We provide all the possible spectral homogenized problems depending on the relations between $\varepsilon $ , $r_{\varepsilon }$ and $\beta (\varepsilon )$ , while we address the convergence, as $\varepsilon \to 0$ , of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on $\varSigma $ . New capacity matrices are introduced to define these strange terms.

Hypergeometric SLE: Conformal Markov Characterization and Applications

This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves—Hypergeometric SLE—denoted by $$\mathrm {hSLE}_{\kappa }(\nu )$$ for $$\kappa \in (0,4]$$ and $$\nu <\kappa -6$$. The proof relies crucially on Dubedat’s commutation relation (Commun Pure Appl Math 60(12):1792–1847, 2007) and a uniqueness result proved in Miller and Sheffield (Ann Probab 44(3):1647–1722, 2016). The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or proved to be conformally invariant) in topological rectangles with alternating boundary conditions. We also prove various properties of $$\mathrm {hSLE}_{\kappa }(\nu )$$ with $$\kappa \in (0,8)$$: continuity, reversibility, target-independence, and conditional law characterization. As by-products, we give two applications of these properties. The first one is about the critical Ising interfaces. We prove the convergence of the Ising interface in rectangles with alternating boundary conditions. This result was first proved by Izyurov (Commun Math Phys 337(1):225–252, 2015), and our proof is different. Our method is based on the properties of $$\mathrm {hSLE}$$ and is easy to generalize to more complicated boundary conditions and to other models. The second application is the existence of the so-called pure partition functions of multiple SLEs. Such existence was proved for $$\kappa \in (0,8){\setminus } {\mathbb {Q}}$$ in Kytola and Peltola (Commun Math Phys 346(1):237–292, 2016), and it was later proved for $$\kappa \in (0,4]$$ in Peltola and Wu (Commun. Math. Phys. 366(2):469–536, 2019). We give a new proof of the existence for $$\kappa \in (0,6]$$ using the properties of $$\mathrm {hSLE}$$.

Elliptic Operators in Multidimensional Cylinders with Frequently Alternating Boundary Conditions Along a Given Curve

We consider a selfadjoint elliptic operator in an infinite multidimensional cylinder with the Dirichlet boundary condition which is replaced by the Robin condition on small sets located along a given line on the boundary. The shape and distribution of these sets are arbitrary. The characteristic linear size of these sets is a small parameter of the problem. It is shown that the resolvent of such an operator converges to the resolvent of the homogenized operator, and an estimate for the convergence rate is obtained. The homogenized operator is the same operator, but without alternating boundary conditions. The difference of resolvents is estimated in the norm of bounded operators acting from L2 to $$ {W}_2^1. $$ If the small sets are periodically distributed and have the same shape and the Robin condition is replaced by the Neumann condition, we derive two-sided asymptotic estimates for the lower band functions, which provides the possibility to estimate from below the length of the first band of the spectrum.

Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity

We consider a homogenization Winkler–Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler–Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order $$\varepsilon $$ . For fixed $$\varepsilon $$ , the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues $$\{\beta _k^\varepsilon \}_{k=1}^{\infty }$$ as $$\varepsilon \rightarrow 0$$ . We show that $$\beta _k^\varepsilon =O(\varepsilon ^{-1})$$ for each fixed k, and we observe a common limit point for all the rescaled eigenvalues $$\varepsilon \beta _k^\varepsilon $$ while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain “groups” of eigenmodes.

Quantum cavities with alternating boundary conditions

We consider the quantum dynamics of a free nonrelativistic particle moving in a cavity and we analyze the effect of a rapid switching between two different boundary conditions. We show that this procedure induces, in the limit of infinitely frequent switchings, a new effective dynamics in the cavity related to a novel boundary condition. We obtain a dynamical composition law for boundary conditions which gives the emerging boundary condition in terms of the two initial ones.

On the uniqueness of global multiple SLEs

This article focuses on the characterization of global multiple Schramm–Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising, FK-Ising and percolation models.

Conformal field theory of critical Casimir forces between surfaces with alternating boundary conditions in two dimensions

Systems as diverse as binary mixtures and inclusions in biological membranes, and many more, can be described effectively by interacting spins. When the critical fluctuations in these systems are constrained by boundary conditions, critical Casimir forces (CCF) emerge. Here we analyze CCF between boundaries with alternating boundary conditions in two dimensions, employing conformal field theory (CFT). After presenting the concept of boundary changing operators, we specifically consider two different boundary configurations for a strip of critical Ising spins: (I) alternating equi-sized domains of up and down spins on both sides of the strip, with a possible lateral shift, and (II) alternating domains of up and down spins of different size on one side and homogeneously fixed spins on the other side of the strip. Asymptotic results for the CCF at small and large distances are derived. We introduce a novel modified Szegö formula for determinants of real antisymmetric block Toeplitz matrices to obtain the exact CCF and the corresponding scaling functions at all distances. We demonstrate the existence of a surface renormalization group flow between universal force amplitudes of different magnitude and sign. The Casimir force can vanish at a stable equilibrium position that can be controlled by parameters of the boundary conditions. Lateral Casimir forces assume a universal simple cosine form at large separations.

Numerical simulation of thermal mass enhanced envelopes for office buildings in subtropical climate zones

The morning power overloading for office cooling is often the aftereffect of not using air conditioning on hot summer nights. Integration of phase change material (PCM) into the office building envelope can be a solution, but, prior to usage, the PCM layer has to be optimized to deal with a number of alternating boundary conditions. The diurnal alternation of the heat load level facing the building envelope is very critical in predicting PCM behavior and cooling energy consumption in a real PCM integrated office building in Hong Kong.EnergyPlus numerical simulation of a PCM layer with specific properties and encased in between two layers of a building envelope was adopted to assess PCM efficiency in relation to its placement, thickness and orientation to the sun, causing moderate and elevated heat loads. The optimized PCM envelope with windows was simulated as a part of a realistic office building model to estimate the air conditioning power balancing on the hottest days and during the entire hot season.The results show that the considered PCM layer should be placed at the inner building envelope part, i.e., close to the office interior. The optimization stage revealed the negative effect of the excessive PCM thermal mass bringing indoor overheating in certain combinations of diurnal heat loads. This is an outcome of not using air conditioning at nights and on Sundays. In the realistic office building model, the negative PCM effect diminishes, though the PCM reduces the morning peak power by up to about 5%. The PCM impact is clearly shown in the comprehensive results for the entire cooling season, where effective smoothing of the AC power demand is indicated. The PCM usage, however, considerably prolongs the AC operation in a narrow power mode (200–244kW) that can aid the chiller selection.The study demonstrates that the morning peak power can be effectively managed via the PCM optimized building envelope in subtropical climate zones. For an office building, a combined window and building envelope with PCM can be an excellent alternative to fully glazed curtain walls.

On the Resolvent of Multidimensional Operators with Frequently Alternating Boundary Conditions with the Robin Homogenized Condition

We consider an elliptic operator in a multidimensional domain with frequent alternation of the Dirichlet condition and the Robin boundary condition in the case where the homogenized operator contains only the original Robin boundary condition. We prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and obtain order sharp estimates for the rate of convergence. We construct a complete asymptotic expansion for the resolvent in the case where the resolvent acts on sufficiently smooth functions and the alternation of boundary conditions is strictly periodic and is given on a multidimensional hyperplane. Bibliography: 23 titles.

Сдвиговые волны в упругом периодическом волноводе с альтернативнымн граничными условиями.

The propagation of shear waves in elastic waveguide of periodic structure consisting of three different materials with alternating along the guide walls boundary conditions is investigated. Using the transfer matrix approach the problem is reduced to the solution of a block transfer matrix eigenvalue problem. Bloth the dispersion and the band gap structure analysis have been carried out numerically. It is shown that for alternating boundary conditions along the waveguide walls, by modulating the ratio of the length of the unit cell to the width of the waveguide, the minimum widths of the stop bands can be moved to the middle of the Brillouin zone

Shear Bloch waves and coupled phonon–polariton in periodic piezoelectric waveguides

Coupled electro-elastic SH waves propagating in a periodic piezoelectric finite-width waveguide are considered in the framework of the full system of Maxwell’s electrodynamic equations. We investigate Bloch–Floquet waves under homogeneous or alternating boundary conditions for the elastic and electromagnetic fields along the guide walls. Zero frequency stop bands, trapped modes as well as some anomalous features due to piezoelectricity are identified. For mixed boundary conditions, by modulating the ratio of the length of the unit cell to the width of the waveguide, the minimum widths of the stop bands can be moved to the middle of the Brillouin zone. The dispersion equation has been investigated also for phonon–polariton band gaps. It is shown that for waveguides at acoustic frequencies, acousto-optic coupling gives rise to polariton behavior at wavelengths much larger than the length of the unit cell but at optical frequencies polariton resonance occurs at wavelengths comparable with the period of the waveguide.

Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures.

Proof of the Razumov–Stroganov conjecture

The Razumov–Stroganov conjecture relates the ground-state coefficients in the periodic even-length dense O ( 1 ) loop model to the enumeration of fully-packed loop configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. Here we prove this conjecture, by means of purely combinatorial methods. The main ingredient is a generalization of the Wieland proof technique for the dihedral symmetry of these classes, based on the ‘gyration’ operation, whose full strength we will investigate in a companion paper.

On a waveguide with an infinite number of small windows

We consider a waveguide modeled by the Laplacian in a straight planar strip with the Dirichlet condition on the upper boundary, while on the lower one we impose periodically alternating boundary conditions with a small period. We study the case when the homogenization leads us to the Neumann boundary condition on the lower boundary. We establish the uniform resolvent convergence and provide the estimates for the rate of convergence. We construct the two-terms asymptotics for the first band functions of the perturbed operator and also the complete two-parametric asymptotic expansion for the bottom of its spectrum.