THE REMOVABLE SINGULARITY THEOREM FOR HARMONIC FUNCTION ON A TWO-DIMENSIONAL STRATIFIED SET

2016 ◽  
Vol 21 (1) ◽  
pp. 108-116
Author(s):  
D.V. Savasteev ◽  
Keyword(s):  
Harmonic Function ◽  
Removable Singularity ◽  
Two Dimensional ◽  
Singularity Theorem ◽  
Stratified Set

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.

The Exact Deformation Theory of Two-Dimensional Dodecagonal Quasicrystal Shaft

2011 ◽  
Vol 213 ◽  
pp. 276-280
Author(s):  
Bao Sheng Zhao ◽  
Ying Tao Zhao ◽  
Yang Gao
Keyword(s):  
Harmonic Function ◽  
Deformation Theory ◽  
Ad Hoc ◽  
Reverse Direction ◽  
Refined Theory ◽  
Two Dimensional ◽  
Surface Loading ◽  
Circular Shaft ◽  
Cylindrical Coordinate ◽  
Classical Elasticity Theory

Cheng’s refined theory is extended to investigate torsional circular shaft of two-dimensional dodecagonal quasicrystal (2D dodecagonal QCs), and Lur’e method about harmonic function is extended to harmonic function in the respective cylindrical coordinate. The exact deformation of torsional circular shaft of 2D dodecagonal QCs under reverse direction surface loading is proposed on the basis of the classical elasticity theory and stress-displacement relations of 2D dodecagonal QCs, and the exact deformation theory provides the solutions about torsional deformation of a circular shaft without ad hoc assumptions. Exact solutions are obtained for circular shaft from boundary conditions. Using Taylor series of the Bessel functions and then dropping all the terms associated with the higher-order terms, we obtain the approximate expressions for circular shaft of 2D dodecagonal QCs under reverse direction surface. To illustrate the application of the theory developed, one example is examined.

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

Note on the slow motion of fluid

1936 ◽  
Vol 32 (4) ◽  
pp. 598-613 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Harmonic Function ◽  
Viscous Liquid ◽  
Stream Function ◽  
Slow Motion ◽  
Normal Derivative ◽  
Great Distance ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Shearing Motion

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

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.

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