Compact Boundary Research Articles

Time asymptotic behavior of the solution to the linear Boltzmann equation in finite bodies

In this paper we discuss time asymptotic behavior of the solution to the Cauchy problem governed by the transport operator in bounded geometry in the case where the boundary conditions are dissipative and modeled by the bounce-back boundary operator plus a compact in $$L^1$$ -spaces. The case of multiplying compact boundary operator is considered in the last subsection.

Some scalar curvature warped product splitting theorems

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata's equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass.

Harmonic Maps and Torse-Forming Vector Fields

In this paper, we prove that any harmonic map from a compact orientable Riemannian manifoldwithout boundary (or from complete Riemannian manifold) (M, g) to Riemannian manifold (N, h)is necessarily constant, with (N, h) admitting a torse-forming vector field satisfying some condition.

QUASICONFORMAL HARMONIC MAPPINGS BETWEEN DOMAINS CONTAINING INFINITY

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

The Effectiveness of Reinforcement and Processing on Mechanical Properties, Wear Behavior and Damping Response of Aluminum Matrix Composites

Abstract Metal matrix composites are an essential product used in engineering materials. This product has wide applications in automotive, aerospace, and other uses because of their lower density, good specific strength, best machinability, and better mechanical properties compared to Al 6063 alloy. In this paper, four different reinforced such as (MA), (MCA), (NFC) and (SA) with a constant rate of 5 wt.% for each reinforced element used and mixed with Al 6063 alloy as a metal matrix composite by using modified two-step mechanical stirrer and having three blades at each step. Coated carbide tool insert was carrying out the turning process. Surface roughness measured after turning of every change in cutting speed. Average chip length and its shape style formation performed within cutting speeds of 10 and 90 m/min, which appears in different length and shapes. Mechanical properties, damping characteristics, and wear loss improved dramatically by adding all reinforced composites to the base Al 6063 alloys. As the results of experiments, the surface roughness decreased by adding all four types of reinforcements. The mechanical properties, wear loss and damping characteristics improved by the constituents of all kinds of reinforcements and also by mixing of all types of constituents together. The effects of MA and NFC are more compared to MCA and SA for improving all conditions of experimental results. Microstructure observation produces compact grain boundaries with strong grains of metal matrix composites compared to Al 6063 alloys.

Semiclassical Resolvent Estimates for $$L^\infty $$ Potentials on Riemannian Manifolds

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.

On the Mixed Steklov—Neumann and Steklov-type Biharmonic Problems in Unbounded Domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Steklov–Neumann and Steklov-type biharmonic problems in unbounded domains with a compact and non-compact boundaries (in particular, the exterior of a compact set, half-space, the domain with conical points) under the assumption that generalized solutions of these problems has a bounded Dirichlet (energy) integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

AbstractWhile it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb {R}^{n+1}$, we show that they must all obey a general “bi-halfspace theorem” (aka “wedge theorem”): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (possibly with compact boundary) $n$-dimensional mean curvature flow self-translating solitons $\Sigma ^n$ in ${\mathbb {R}}^{n+1}$ up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman–Meeks in 1989 for minimal submanifolds: all of ${\mathbb {R}}^{n}$, halfspaces, slabs, hyperplanes, and convex compacts in ${\mathbb {R}}^{n}$.

$L^p$ Hardy inequality on $C^{1,\gamma}$ domains

We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,\gamma}$ with $\gamma \in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for $C^2$-domains.

Reinforcement and processing on the machinability and mechanical properties of aluminum matrix composites

Metal matrix composites are an essential product used in engineering materials. This product has wide applications in automotive, aerospace, and other uses. In this paper, four different reinforced such as (MA), (MCA), (NFC) and (SA) with a constant rate of 5wt. % for each reinforced element used and mixed with Al 6063 alloy to produced composite by using modified two-step mechanical stirrer and having three blades at each step. Coated carbide tool insert was carrying out the turning process. The chip volume ratio and chip shape style formation were performed within cutting speeds of 10 and 90m/min, which appears in different length and shapes especially at 90m/min. The chip volume ratio is not increased by all four types of reinforced metal but still maintain in the standard range according to the shape of chips. NFC increasing average chip length during cutting speed of 10m/min, but within cutting speed of 90m/min, MA has an important role to change the volume and formation style of a chip from discontinuous chip to continuous chip. It should be noted from the results, that the NFC has decreased surface roughness dramatically for all cutting speeds and followed by MCA, but the MA and SA have less effect compared to NFC and MCA. The mechanical properties such as yield ultimate tensile strength and hardness of the composites are much more than Al 6063 alloy especially when adding all four types of reinforcement together moreover the effect of NFC is very less compared with other types of reinforced materials. MA increase hardness more than the different types of reinforcement. Microstructure observation produces compact grain boundaries with strong grains of metal matrix composites compared to Al 6063 alloys.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah–Singer Dirac operator / D B in L 2 depends Riesz continuously on L ∞ perturbations of local boundary conditions B . The Lipschitz bound for the map B → / D B ( 1 + / D B 2 ) − 1 2 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Vanishing theorems for the cohomology groups of free boundary submanifolds

In this paper, we prove that there exists a universal constant C, depending only on positive integers $$n\ge 3$$ and $$p\le n-1$$ , such that if $$M^n$$ is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball $$\mathbb {B}^{n+k}$$ whose size of the traceless second fundamental form is less than C, then the pth cohomology group of $$M^n$$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $$\mathbb {B}^{2+k}$$ .

Synergistic Effects Between the Two Choline-Based Ionic Liquids as Lubricant Additives in Glycerol Aqueous Solution

Two choline-based ionic liquids (ILs) were used as lubricant additives in the aqueous glycerol solution, and the tribological properties of their different combinations were investigated. They exhibit remarkable synergistic effects on the antiwear and friction-reducing abilities. The combinations of both ILs more significantly improve the lubricity of the glycerol solution compared with the independent additives. In addition, the tribological behavior of the optimal combination was explored under different temperature and load conditions. The lubricating property of the optimal additive formulation is weakened with the increase of temperature, but enhanced with the increase of load. The worn surfaces were analyzed using SEM-EDAX and XPS. Meanwhile, Monte Carlo simulations were performed to study the adsorption behavior of the additives on iron surface in the presence of water. Remarkable synergistic lubricating interactions between the ILs are ascribed to the formation of compact and stable boundary films composed of mixed adsorbed layers and complex tribofilms.

On the Gevrey well-posedness of the Kirchhoff equation

This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $\gamma^s_{\eta,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains and in exterior domains with compact boundary.

Harnack Inequalities for Simple Heat Equations on Riemannian Manifolds

In this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem $$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: $$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $$ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).

S-shaped bifurcation diagrams in exterior domains

We study a nonlinear eigenvalue problem on the exterior to a simply connected bounded domain in \(\mathbb {R}^N\) containing the origin. We consider positive weak solutions satisfying Dirichlet boundary conditions on the compact boundary and decaying to zero at infinity. We discuss multiplicity and uniqueness results of solutions with respect to a bifurcation parameter and conjecture an S-shaped bifurcation diagram for positive reaction terms which are singular at the origin and sublinear at infinity. As a by-product, on regions exterior to a ball with radially symmetric weight functions, we obtain radial symmetry of solutions when uniqueness holds.

Attempts at vacuum counterexamples to cosmic censorship in AdS

We consider vacuum solutions of four dimensional general relativity with Λ < 0. We numerically construct stationary solutions that asymptotically approach a boundary metric with differential rotation. Smooth solutions only exist up to a critical rotation. We thus argue that increasing the differential rotation by a finite amount will cause the curvature to grow without bound. This holds for both zero and nonzero temperature, and both compact and noncompact boundaries. However, the boundary metric always develops an ergoregion before reaching the critical rotation, which probably means that the energy is unbounded from below for these counterexamples to cosmic censorship.

Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $\Sigma$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-d\sigma_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then $(S,g_1)$ is parabolic.

The Effect of Grain Boundary Structure on the Intergranular Degradation Behavior of Solution Annealed Alloy 690 in Simulated Pressurized Water Reactor Primary Environment

The environmental degradation of four different types of grain boundaries were investigated on alloy 690 following slow strain rate tensile tests in 360 oC hydrogenated water. Random high angle boundaries (RHABs) support fast Cr diffusion that promotes the formation of a compact surface oxide film and grain boundary migration. Surprisingly, coherent twin boundaries (CTBs) are susceptible to intergranular oxidation and do not exhibit Cr diffusion or grain boundary migration. When CTBs are changed to transformed twin boundaries (TTBs) by cold work, they behave like RHABs. Finally, incoherent twin boundaries (ITBs) undergo intergranular oxidation with limited Cr depletion but no boundary migration beyond the oxide. The Cr diffusivity along grain boundary in this alloy is directly related to the density of coincident site in the grain boundary plane and determines the morphology of oxide formed near the grain boundary. CTBs are still highly resistant to stress corrosion cracking (SCC) due to the semi-coherent interface between the intergranular chromia and grain matrix. In contrast, the intergranular oxides formed along RHABs inherit highly-disordered boundary structure from the original grain boundaries and show much higher SCC susceptibility. The grain boundary structure dependence of SCC resistance should be understood from its effects on solute diffusivity, structure of intergranular oxide and the local stress-strain state.

An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.