Removable Singularities Research Articles

On removable singularities of harmonic functions on a stratified set

We consider sets removable for bounded harmonic functions on a stratified set with flat interior strata. We establish that relatively closed sets of finite Hausdorff (n-2)-measure are removable for bounded harmonic functions on an n-dimensional stratified set satisfying the “strong sturdiness” condition.

On Removable Singularities of Harmonic Functions on a Stratified Set

On Removable Singularities of Harmonic Functions on a Stratified Set

Removable singularities in the boundary for quasilinear elliptic equations

Removable singularities in the boundary for quasilinear elliptic equations

Removable Singularities of Harmonic Functions on Stratified Sets

Removable Singularities of Harmonic Functions on Stratified Sets

Removability of time-dependent singularities of solutions to the Navier-Stokes equations

Let Ω be a bounded domain in RN and ξ∈Cα([0,T];Ω) for 1/N<α≤1. Suppose that u is a smooth solution of the Navier-Stokes equations in ⋃0<t<T(Ω∖{ξ(t)})×{t}, namely, ⋃0<t<T{ξ(t)}×{t} is supposed to be the set of moving singularities of u in Ω×[0,T]. We prove that ifu(x,t)=o(|x−ξ(t)|−N+β)locally uniformly in t∈(0,T) as x→ξ(t) for β≡max⁡{1/α,N−1}, then u is, in fact, smooth in the whole region Ω×(0,T). Our result may be regarded as a theorem on removable time-dependent singularities of solutions to the Navier-Stokes equations.

Albanese Maps for Open Algebraic Spaces

Abstract We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the structural properties of group schemes, removal of singularities by alterations, and ind-objects. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields.

Listening to quantum gravity?

Recent experimental progresses in controlling classical and quantum fluids have made it possible to realize acoustic analogs of gravitational black holes, where a flowing fluid provides an effective spacetime on which sound waves propagate, demonstrating Hawking-like radiation and superradiance. We propose the exciting possibility that new hydrodynamic systems might provide insights to help resolve mysteries associated with quantum gravity, including the black hole information-loss paradox and the removal of spacetime singularities.

Unified analysis of finite-size error for periodic Hartree-Fock and second order Møller-Plesset perturbation theory

Despite decades of practice, finite-size errors in many widely used electronic structure theories for periodic systems remain poorly understood. For periodic systems using a general Monkhorst-Pack grid, there has been no comprehensive and rigorous analysis of the finite-size error in the Hartree-Fock theory (HF) and the second order Møller-Plesset perturbation theory (MP2), which are the simplest wavefunction based method, and the simplest post-Hartree-Fock method, respectively. Such calculations can be viewed as a multi-dimensional integral discretized with certain trapezoidal rules. Due to the Coulomb singularity, the integrand has many points of discontinuity in general, and standard error analysis based on the Euler-Maclaurin formula gives overly pessimistic results. The lack of analytic understanding of finite-size errors also impedes the development of effective finite-size correction schemes. We propose a unified analysis to obtain sharp convergence rates of finite-size errors for the periodic HF and MP2 theories. Our main technical advancement is a generalization of the result of Lyness [Math. Comp. 30 (1976), pp. 1–23] for obtaining sharp convergence rates of the trapezoidal rule for a class of non-smooth integrands. Our result is applicable to three-dimensional bulk systems as well as low dimensional systems (such as nanowires and 2D materials). Our unified analysis also allows us to prove the effectiveness of the Madelung-constant correction to the Fock exchange energy, and the effectiveness of a recently proposed staggered mesh method for periodic MP2 calculations (see X. Xing, X. Li, and L. Lin [J. Chem. Theory Comput. 17 (2021), pp. 4733–4745]). Our analysis connects the effectiveness of the staggered mesh method with integrands with removable singularities, and suggests a new staggered mesh method for reducing finite-size errors of periodic HF calculations.

Hopf-like bifurcation in a wave equation at a removable singularity

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.

Removable Singularities for Solutions of the Fractional Heat Equation in Time Varying Domains

In this paper we study removable singularities for solutions of the fractional heat equation in the spacial-time space. We introduce associated capacities and we study some of its metric and geometric properties.

Removable singularities for subsolutions of elliptic equations involving weighted variable exponent Sobolev spaces

Removable singularities for subsolutions of elliptic equations involving weighted variable exponent Sobolev spaces

On removable singular sets for solutions of higher order differential inequalities

We obtain conditions guaranteeing that weak solutions of higher order differential inequalities have a removable singularity on a compact set. These conditions depend on the fractal dimension of the singular set. For solutions of the nonlinear Poisson equation in the case of an isolated singularity, i.e. in the case where the fractal dimension of the singular set is equal to zero, they coincide with the well-known Brezis-Véron condition.

Analysis of interaction of hydraulic fractures with natural fractures and bedding planes in layered formation through cohesive zone modelling

A model capturing micro-void nucleation with tensile and shear failure modes is developed to present interaction mechanism of propagating hydraulic fracture(s) (HF(s)) with shale bedding planes (BPs) and natural fracture(s) (NF(s)). The study objective is to translate algorithm for excess shear forces into understanding of NF and BP slip and dilation at junction of their intersection with HFs. The methodology is based on regularized global cohesive zone method (CZM) using viscous regularization implemented with Mohr- Coulomb frictional shear plasticity implemented through Newton-Raphson return mapping algorithm. The approach leverages understanding of multiple fracture geometries (width, length, height) when HF approaches NFs and BPs. In situ stress contrast shows more influence on creation of fracture complexity. Cluster spacing is shown to have moderate influence on shear opening at the intersection of HFs with BPS and NFs. Low dipping and weak shale beds demonstrated with dip angles of 30° and 45° and bed interfacial material strength of 1 MPa tend to propagate HFs along BPs. A 90 dip develops T-shaped fractures at weak interfacial strength. Comparison of field measured bottom-hole pressure (BHP) with simulated BHP in layered shale reservoir in the western Sichuan basin gives maximum error as low as 6 %. A measurement of junction dilation of 0.0055 m was recorded at highest injection rate 0.04 m3/s with an observation that the dilation progressively increased as stimulation time was prolonged. On contrary, lower injection rates of 0.003 m3/s, 0.01 m3/s and 0.02 m3/s recorded maximum dilation before the end of total stimulation time. Three-point bending test based on published experiment on shale play is used to test the performance of this model. Comparison with available solution in literature indicates superiority of the developed CZM vested in its removal of stress singularities at the fracture tips replacing it with the development of fracture process zone softening effect and micro-crack nucleation. This study is important in understanding field implementation of HFs in formations with NFs/BPs such as Longmaxi shale gas reservoir.

Removable time-dependent singularities of solutions to the Stokes equations

Let Ω⊂RN and let ξ∈Cα([0,T];Ω) for 0<α≤12. We consider the situation that u=u(x,t) is a classical solution of the Stokes equations in ⋃0<t<T(Ω∖{ξ(t)})×{t}, that is, {ξ(t)}0<t<T is regarded as the time-dependent singularities of u in Ω×(0,T). If u behaves around ξ(t) like |u(x,t)|=o(|x−ξ(t)|2−N+(1/α−2)) as x→ξ(t) uniformly in t∈(0,T), then {ξ(t)}0<t<T is a family of removable singularities of u, which implies that u can be extended as a smooth solution in the whole space and time Ω×(0,T).

On removable singularities for solutions of Neumann problem for elliptic equations involving variable exponent

We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this singularity for equations in the variable exponent Sobolev space $$W^{1,p(\cdot )}(\Omega )$$ .

Removal of conical singularities from rotating C-metrics and dual CFT entropy

We show how to remove from the rotating C-metric spacetime, which describes accelerating Kerr black holes, both conical singularities. This can be done by embedding the metric into a swirling gravitational universe, through a proper Ehlers transformation. The spin-spin interaction between the external rotational background and the black hole provides the source of the acceleration without the need of rods or strings. The physical properties and entropy of the new solution are studied using near horizon and dual conformal techniques of the Kerr/CFT correspondence. The charged case is also analysed: accelerating Reissner-Nordstrom and Accelerating Kerr-Newman space-times embedded in a swirling universe are also generated.

Removable singularity of positive mass theorem with continuous metrics

In this paper, we consider asymptotically flat Riemannian manifolds endowed with a continuous metric and the metric is smooth away from a compact subset with certain conditions. If the metric is Lipschitz and the scalar curvature is nonnegative away from a closed subset with \((n-1)\)-dimensional Hausdorff measure zero, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that the manifold is isometric to the Euclidean space. The Hausdorff measure condition is optimal, which confirms a conjecture of Lee in 2013. The proof of the nonnegativity is based on a smoothing argument with some new estimates such as a \(L^1\)-scalar curvature approximation estimates. The proof of the rigidity relies on the RCD theory where we show that the manifold has nonnegative Ricci curvature in RCD sense. The idea involving RCD theory in the proof turns out to be much more suitable in the study of the rigidity for continuous metrics. Furthermore, our result generalizes the result of Lee and LeFloch (Commun Math Phys 339(1):99–120, 2015) from spin to non-spin.

The analysis of inhomogeneous Yang–Mills connections on closed Riemannian manifold

In this article, we study a class of connections on a closed Riemannian manifold X of dimensional n &amp;gt; 4, which we call inhomogeneous Yang–Mills connections. Some special cases included Ω-Yang–Mills connections, where Ω is a smooth (may be not closed) (n − 4)-form on X. We extend the known analytic results of pure Yang–Mills connections, which included the monotonicity formula and the ɛ-regularity theorem to the class of inhomogeneous Yang–Mills connections. Using those analytic results, we obtain the energy quantization and Uhlenbeck compactness for the moduli space of inhomogeneous Yang–Mills connections that have a uniformly Ln2-bounded curvature. A removable singularity theorem for singular inhomogeneous Yang–Mills connections on a bundle over the punctured ball is also proved. Finally, we also prove an energy gap result for inhomogeneous Yang–Mills connections under some mild conditions.

Removable singularities for bounded [formula omitted]-(super)harmonic and quasi(super)harmonic functions on weighted [formula omitted

It is well known that sets of p-capacity zero are removable for bounded p-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted Rn when n>p, although only in very special cases. A complete characterization of removable singularities for bounded A-harmonic functions on weighted Rn, n≥1, is also given, where the weight is p-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted Rn, n≥2, as well as on unweighted R. For bounded A-superharmonic functions and bounded quasisuperharmonic functions on weighted Rn, n≥2, we show that relatively closed sets are removable if and only if they have zero capacity.

Elastic and plastic fields induced by a screw dislocation in a nanowire within Mindlin’s second strain gradient theory

Due to the wide applications of nanowires as a main component of nano-scale electronic devices, a lot of attentions have thus far been paid to study their relevant problems. Knowing that the presence of dislocations in nanowires affects their mechanical properties, this paper is dedicated to the determination of analytical expressions for the elastic displacement, elastic strains, and plastic distortion developed in a nanowire due to a screw dislocation, as well as the image force acting on the dislocation core, within the framework of Mindlin’s second strain gradient theory. Expecting that the size effect plays an important role in such a problem, this theory seems an appropriate continuum approach for its addressing. Furthermore, utilizing the adopted theory leads to the removal of all classical singularities in the elastic displacement, strain, stress, and plastic distortion fields, as well as the image force. It is, in addition, seen that the elastic double- and elastic triple-strains are regularized. Furthermore, the results evidently illustrate the size effect, i.e. the effect of the ratio between the radius of the nanowire and the characteristic lengths of its constituent material.