scholarly journals A Removable Singularity Theorem for Local Homeomorphisms

1970 ◽  
Vol 20 (5) ◽  
pp. 455-461 ◽  
Author(s):  
Stephen Agard ◽  
Albert Marden
Keyword(s):  
Removable Singularity ◽  
Singularity Theorem

REGULARITY OF WEAK SOLUTIONS FOR THE NAVIER–STOKES EQUATIONS IN THE CLASS L∞(BMO-1)

2012 ◽  
Vol 14 (03) ◽  
pp. 1250020 ◽  
Author(s):  
WENDONG WANG ◽  
ZHIFEI ZHANG
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Removable Singularity ◽  
Navier Stokes ◽  
Singularity Theorem ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Mild Assumption

We study the regularity of weak solution for the Navier–Stokes equations in the class L∞( BMO-1). It is proved that the weak solution in L∞( BMO-1) is regular if it satisfies a mild assumption on the vorticity direction, or it is axisymmetric. A removable singularity theorem in ∈ L∞( VMO-1) is also proved.

scholarly journals Isolated Singularities of Yang–Mills–Higgs Fields on Surfaces

2020 ◽  
Author(s):  
Bo Chen ◽  
Chong Song
Keyword(s):  
Decay Estimate ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Removable Singularity ◽  
Fiber Space ◽  
Singularity Theorem ◽  
Isolated Singularities ◽  
Asymptotic Decay ◽  
Yang Mills ◽  
Higgs Fields

Abstract We study isolated singularities of 2D Yang–Mills–Higgs (YMH) fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general, the singularity cannot be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the YMH field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of 2D harmonic maps.

scholarly journals Some Properties of Solutions to Weakly Hypoelliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Christian Bär
Keyword(s):  
Parabolic Equations ◽  
Complex Analysis ◽  
Bounded Solution ◽  
Removable Singularity ◽  
Local Solution ◽  
Singularity Theorem ◽  
Hypoelliptic Equation ◽  
Constant Coefficients ◽  
Hypoelliptic Equations ◽  
Liouville’S Theorem

A linear different operatorLis called weakly hypoelliptic if any local solutionuofLu=0is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and anyLp-solution must vanish.

scholarly journals Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

2018 ◽  
Vol 8 (1) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Sunghan Kim ◽  
Henrik Shahgholian
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonnegative Solution ◽  
Removable Singularity ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Isolated Singularity ◽  
Semilinear Elliptic

Abstract We study the semilinear elliptic equation -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where {B_{1}\subset{\mathbb{R}}^{n}} , with {n\geq 3} , {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution {u\in C^{2}(B_{1}\setminus\{0\})} of the above equation either has a removable singularity at the origin or it behaves like u(x)=A(1+o(1))|x|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}

scholarly journals A NEW SINGULARITY THEOREM IN RELATIVISTIC COSMOLOGY

2000 ◽  
Vol 15 (06) ◽  
pp. 391-395 ◽  
Author(s):  
A. K. RAYCHAUDHURI
Keyword(s):  
Ricci Tensor ◽  
Energy Condition ◽  
Space Time ◽  
Relativistic Cosmology ◽  
Raychaudhuri Equation ◽  
Singularity Theorem ◽  
Strong Energy Condition ◽  
Strong Energy

It is shown that if the time-like eigenvector of the Ricci tensor is hypersurface orthogonal so that the space–time allows a foliation into space sections, then the space average of each of the scalars that appears in the Raychaudhuri equation vanishes provided that the strong energy condition holds good. This result is presented in the form of a singularity theorem.

scholarly journals A stochastic proof of an extension of a theorem of Rado

1983 ◽  
Vol 26 (3) ◽  
pp. 333-336 ◽  
Author(s):  
Bernt Øksendal
Keyword(s):  
Brownian Motion ◽  
Analytic Function ◽  
Meromorphic Function ◽  
Conformal Invariance ◽  
Analytic Functions ◽  
Removable Singularity ◽  
Boundary Behaviour ◽  
Type Result ◽  
Cluster Set ◽  
Asymptotic Values

The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).

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