Compact Boundary Research Articles

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,\cdots, D^mu)dx$ as in (1.1). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincare-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [19] with some techniques from [27,43,46]. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.

Subgrade Compaction Boundary Layer Detection Method and Detection System

At present, the degree of compaction is the main criterion for subgrade compaction quality.It is the relative expression of the compacted density,Just average.It does not adequately reflect the subgrade compaction layer vertical compaction density distribution law. Compaction boundary layer micro-unit compression pressure on in the process of pressure transmission decreases gradually until they reach the critical formation pressure dense layer. Compaction by detecting the boundary layer vertical zone layer densification, to calculate the boundary thickness, to draw isodense of densification. a clear reproduction of the compacted layer vertical compaction density distribution law. This paper describes the detection principles and detection methods of the compaction boundary layer. describes in detail functional structure and system design of the vehicle automatic detection system used to detect compaction boundary layer of each vertical zone.

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

AbstractIn this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Convex spacelike hypersurfaces of constant curvature in de Sitter space

We show that for a very general and natural class of curvature functions (for example the curvature quotients$(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurfacein de Sitter space satisfying$f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinityhas at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.

Partition functions on the Euclidean plane with compact boundaries in conformal and non-conformal theories

In this paper we calculate the exact partition function for free bosons on the plane with lacunae. First the partition function for a plane with two spherical holes is calculated by matching exactly for the infinite set of Wilson coefficients in an effective world line theory and then performing the ensuing Gaussian integration. The partition is then re-calculated using conformal field theory techniques, and the equality of the two results is made manifest. It is then demonstrated that there is an exact correspondence between the Wilson coefficients (susceptibilities) in the effective field theory and the weights of the individual excitations of the closed string coherent state on the boundary. We calculate the partition function for the case of three holes where CFT techniques necessitate a closed form for the map from the corresponding closed string pants diagrams. Finally, it is shown that the Wilson coefficients for the case of quartic and higher order kernels, where standard CFT techniques are no longer applicable, can also be completely determined. These techniques can also be applied to the case of non-trivial central charges.

Dynamics of Compressible Non-isothermal Fluids of Non-Newtonian Korteweg Type

The equations of motion for compressible fluids of Korteweg type as derived by Dunn and Serrin in 1985 are studied in their full generality: the Korteweg tensor is assumed to be an arbitrary function of the form $\mathcal{K} := \left( - \rho^2 \partial_{\rho} \psi + \rho \nabla \cdot ( \kappa \nabla \rho) \right) \mathcal{I} - \kappa \nabla \rho \otimes \nabla \rho, \quad \kappa := 2 \rho \partial_{\phi} \psi(\rho,\theta,\phi), \quad \phi:=|\nabla \rho|^2,$ where $\psi$ denotes Helmholtz free energy density and the capillarity $\kappa$ is subject only to the natural positivity conditions $\kappa(\rho,\theta,\phi) >0, \quad \kappa(\rho,\theta,\phi) + 2 \phi \partial_{\phi} \kappa(\rho,\theta,\phi) > 0, \quad \rho, \theta, \phi \ge 0.$ The viscous stress is supposed to be of generalized Newtonian type. The main result of the paper establishes well-posedness on domains with compact boundaries; the proof is based on refined methods of maximal regularity.

Solitary waves in critical Abelian gauge theories

We prove existence of standing waves solutionsfor electrostatic Klein-Gordon-Maxwell systems in arbitrary dimensional compact Riemannian manifoldswith boundary for zero Dirichlet boundary conditions.We prove that phase compensation holds true when the dimension $n = 3$ or $4$.In these dimensions, existence of a solution is obtained when the mass of the particle field, balanced by the phase,is small in a geometrically quantified sense. In particular, existence holds true for sufficiently largephases. When $n \ge 5$, existence of a solution is obtained when the mass of the particle field is sufficiently small.

Invariants of the harmonic conformal class of an asymptotically flat manifold

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic conformal class. These invariants are closely related to constrained geometric optimization problems involving hypersurface area-minimizers and the ADM mass. In the final section, we discuss possible applications of the invariants and their relationship with zero area singularities and the positive mass theorem.

A compact minimal shadow boundary in Euclidean space is totally geodesic

We prove that a compact minimal shadow boundary of a hypersurface in Euclidean space is totally geodesic. We show that shadow boundaries detect principal directions and umbilical points of a hypersurface. As application we deduce that every shadow boundary of a compact strictly convex surface contains at least two principal directions.

Harmonic mappings and distance function

We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with C1,α (α < 1) and, respectively, C1,1 compact boundary is bi-Lipschitz. This theorem extends a similar result of the author [10] for Jordan domains, where stronger boundary conditions for the image domain were needed. The proof uses distance function from the boundary of the image domain. Mathematics Subject Classification (2010): 58E20 (primary); 30C62 (secondary).

Bonded hydrogen in nanocrystalline silicon photovoltaic materials: Impact on structure and defect density

We have performed a detailed structural and optical investigation of hydrogenated nanocrystalline silicon (nc-Si:H) thin films prepared by plasma-enhanced chemical vapor deposition. The microstructural properties of these thin films are characterized and interpreted physically based on the growth mechanism. Infrared spectroscopy reveals that the bonded hydrogen in a platelet-like configuration, which is believed to be located at grain boundaries, greatly affects oxygen incursions into nc-Si:H thin films, whereas electron spin resonance observations link these incursions to the introduction of dangling bond defects. Consequently, we propose that in nc-Si:H thin films, high bonded-hydrogen content in grain boundaries is of great importance in forming hydrogen-dense amorphous tissues around the small crystalline grains, i.e., compact grain boundary structures with good passivation. Such structures effectively prevent post-deposition oxidation of grain boundary surfaces, which might lead to the formation of dangling bond defects.

Proper shape over finite coverings

A proper shape is presented using an intrinsic definition and only finite coverings of open sets with compact boundary. For locally compact separable metric spaces with compact spaces of quasicomponents, it is shown: If X and Y have the same proper shape over finite coverings then their end points compactifications FX and FY have the same shape.

General solution for Eshelby’s problem of 2D arbitrarily shaped piezoelectric inclusions

Eshelby’s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media. For example, it intervenes in the nanostructure design of strained semiconductor devices involving strain-induced quantum dot (QD) and quantum wire (QWR) growth. Using the extended Stroh formalism, the present work gives a general analytical solution for Eshelby’s problem of two-dimensional arbitrarily shaped piezoelectric inclusions. The key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended Stroh formalism applied to Eshelby’s problem. The simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle Eshelby’s piezoelectric problem for a large variety of non-elliptical inclusions. In the present work, explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by Jordan’s curves and Laurent’s polynomials. By considering the piezoelectric material GaAs (110), the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions, and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion.

Circle decompositions of surfaces

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.

Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system Δ u − ∂ u ∂ t = λ p ( x , t ) g ( v ) , Δ v − ∂ v ∂ t = μ q ( x , t ) f ( u ) in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on ( 0 , ∞ ) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J ∞ ( D ) .

Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m (−1) α D α a α,β (x)D β , a α,β (·) ∈ C ∞( $$ \bar \Omega $$ ) on smooth (bounded or unbounded) domains Ω in ℝ n with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem Δu = λp(x)f(u) in Ω subject to some Dirichlet conditions, where Ω is a regular domain in ℝn (n ≥ 3) with compact boundary. The nonlinearity f is nonnegative continuous and the potential p belongs to some Kato class K(Ω). So we prove the existence of a positive continuous solution depending on λ by the use of a potential theory approach.

On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions

We study the low Mach number limit for the compressible Navier-Stokes system supplemented with Navier's boundary condition on an unbounded domain with compact boundary. Our main result asserts that the velocities converge pointwise to a solenoidal vector field - a weak solution of the incompressible Navier-Stokes system - while the fluid density becomes constant. The proof is based on a variant of local energy decay property for the underlying acoustic equation established by Kato.

A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in ℝ 3 with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary value problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solution of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

On Levi-flat Hypersurfaces with Prescribed Boundary

We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence of $M$. All CR points have to be nonminimal and all complex points have to be flat. Then, adding a positivity condition at complex points, which is similar to the ellipticity for $n=2$ and excluding the possibility of $S$ to contain complex $(n-2)$-dimensional submanifolds, we obtain a solution $M$ to the above problem as a projection of a possibly singular Levi-flat hypersurface in $R\times C^n$. It turns out that $S$ has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of $M$. There are no more global assumptions on $S$ like being contained in the boundary of a strongly pseudoconvex domain, as it was in case $n=2$. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary $S$ must coincide with the constructed solution.