Removable Singularities Research Articles

Removable singularities of holomorphic functions

Доказываются несколько достаточных условий устранимости особых множеств нулевой площади для голоморфных функций в плоских областях и на римановых поверхностях при некоторых метрических и топологических условиях на предельные множества функций и их графиков. Библиография: 14 названий.

Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term

Let Ω be a domain in R with N ≥ 2 and 0 ∈ Ω. For 0 0 and m+q > 1, we obtain a complete classification of the behaviour near 0 (as well at∞ if Ω = R ) for all positive C(Ω\{0}) solutions of the elliptic equation ∆u = u|∇u| in Ω {0}, together with corresponding existence results. We prove that (a) when Ω = R , any positive solution with a removable singularity at 0 must be constant; (b) If q∗ := N−m (N−1) N−2 for N ≥ 2 and E denotes the fundamental solution of the Laplacian, then for 0 ≤ q < q∗, any positive solution has either a removable singularity at 0, or lim|x|→0 u(x)/E(x) ∈ (0,∞) or lim|x|→0 |x|u(x) = λ with θ and λ uniquely determined positive constants. When Ω = R , we establish that any positive solution is radially symmetric and non-increasing with (possibly any) non-negative limit at ∞. (c) If, in turn, q ≥ q∗ for N ≥ 3, then 0 is a removable singularity for all positive solutions. This is joint work with Joshua Ching (The University of Sydney). School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

Removable Singularities of the cscK Metric

In this paper, we consider a CscK metric defined away from divisor and with metric upper bound and lower bound going to zero in certain rate. And we'll prove that this nicely behaved metric is a smooth CscK metric across the divisor.

Removable singularities for div v=f in weighted Lebesgue spaces

Let $w\in L^1_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p_{1/w}$, $1<p<+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.

On removable singularities of a class of mappings (in Russian)

On removable singularities of a class of mappings (in Russian)

Singularities of discrete open mappings with controlled p-module

We consider the generic discrete open continuous mappings in ℝn under which the perturbation of extremal lengths of curve collections is controlled integrally via ∫ Q(x)ηp(|x−x0|)dm(x) with p ∈ (n−1, n), where Q is a measurable function on ℝn and \(\int_{{r_1}}^{{r_2}} {\eta (r)dr \geqslant 1} \) for any η on a given interval [r1, r2]. The main result is that, in contrast to the canonical classical case of holomorphic functions, the above mappings may only have removable singularities.

Boundary value problems with higher order Lipschitz boundary data for polymonogenic functions in fractal domains

In this note we consider certain jump problem for poly-monogenic functions in fractal domains with higher order Lipschitz boundary data. This is accomplished by using a higher order Teodorescu operator which replaces the expected surface integral. Also, we give out the uniqueness of solutions basing the work on the method of removable singularities for monogenic functions making use of a Dolzhenko type theorem.

A remark on removable singularity for nonlinear convection–diffusion equation

In this paper we study the following Cauchy problem: ut=uxx+(un)x,(x,t)∈R×(0,∞),u(x,0)=δ(x),x∈R, where δ(x) is a Dirac measure and n≥0. Its solution is called source-type solution. Such singular solution plays an important role in the development of theory of nonlinear parabolic equations. However, there seems not to be perfect answer to the research. Here we focus on whether there is a critical exponent n0 such that when n<n0 there exists unique source-type solution, while n≥n0 there is no source-type solution, and on what the singular expansions of source-type solutions at origin are. From a physical point of view, there are phenomena of the interactive effect between the diffusion and convection in a heat process, which is re-confirmed and described through mathematical analysis and numerical simulation. In addition, thanks to the entropy inequality, we get new proof of uniqueness and are able to extend our approaches to nonlinear parabolic-hyperbolic equations with Radon measure as initial datum.

Removable singularities for the Von Karman equations

A theorem for removable singularities of the fourth order system of equations in the plane called the Von Karman equations is established. This result is best possible in several different ways. The key idea is to apply the Calderon and Zygmund singular integral theory in conjunction with several lemmas to a nontrivial bootstrap argument.

Remarks on the Green’s function of the linearized Monge–Ampère operator

In this note, we obtain sharp bounds for the Green’s function of the linearized Monge–Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge–Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman–Stampacchia–Weinberger for uniformly elliptic operators in divergence form. We also obtain the L p integrability for the gradient of the Green’s function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge–Ampère equation.

Numerically stable formulas for a particle-based explicit exponential integrator

Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vector-valued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particle-based (Lagrangian-type) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the X-IVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using double-precision floating-point arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourth-order divided differences of the exponential function.

On removable singularities of maps with growth bounded by a function

This paper studies questions related to the local behavior of almost everywhere differentiable maps with the N, N −1, ACP, and ACP −1 properties whose quasiconformality characteristic satisfies certain growth conditions. It is shown that, if a map of this type grows in a neighborhood of an isolated boundary point no faster than a function of the radius of a ball, then this point is either a removable singular point or a pole of this map.

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

We study qualitative properties of positive singular solutions to a two-coupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrödinger equations with critical exponents for nonlinear optics and Bose-Einstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the same system in a punctured ball will be studied.

Minimizers of Higher Order Gauge Invariant Functionals

We introduce higher order variants of the Yang–Mills functional that involve \((n-2)\)-th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions \(\mathrm {dim}M\le 2n\). These results are then used to establish the existence of smooth minimizers on a given principal bundle \(P\rightarrow M\) for subcritical dimensions \(\mathrm {dim}M<2n\). In the case of critical dimension \(\mathrm {dim}M=2n\) we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes \(c_1,\ldots ,c_{n-1}\). A key result is a removable singularity theorem for bundles carrying a \(W^{n-1,2}\)-connection. This generalizes a recent result by Petrache and Riviere.

Classification of isolated singularities for nonhomogeneous operators in divergence form

Consider the equation div(φ(|∇u|)|∇u|∇u)=0 on the punctured unit ball from RN (N≥2), where φ is an odd, increasing homeomorphism from R onto R of class C1. Under reasonable assumptions on φ we prove that if u is a non-negative solution of our equation, then either 0 is a removable singularity of u or u behaves near 0 as the fundamental solution of the equation investigated here. In particular, our result complements to the case on nonhomogeneous operators in divergence form Bôcher's Theorem and some classical results by Serrin.

Exceptional Sets for Subharmonic Functions

Blanchet has shown that hypersurfaces of class C1 are removable singularities for subharmonic functions, provided the considered subharmonic functions satisfy certain assumptions. Later we showed that, in certain cases, it is sufficient that the exceptional sets are of finite (n-1)-dimensional Hausdorff measure. Now we improve our results still further, relaxing our previous assumptions imposed on the considered subharmonic functions.

A theorem on the removal of boundary singularities of pseudo-holomorphic curves

A theorem on the removal of boundary singularities of pseudo-holomorphic curves

Solutions of the differential inequality with a null Lagrangian: higher integrability and removability of singularities. II

The aim of this paper is to establish a result on removability of singularities for solutions of the differential inequality with a null Lagrangian. Also, we obtain integral estimates for wedge products of closed differential forms and for minors of a Jacobian matrix.

3D viscous incompressible fluid around one thin obstacle

In this article, we consider Leray solutions of the Navier-Stokes equations in the exterior of one obstacle in 3D and we study the asymptotic behavior of these solutions when the obstacle shrinks to a curve or to a surface. In particular, we will prove that a solid curve has no effect on the motion of a viscous fluid, so it is a removable singularity for these equations.

Removable singularities of quasilinear parabolic equations with coefficients from Kato-type classes

We establish the best possible condition for point singularities to be removable for nonlinear parabolic equations in divergent form with lower-order terms from the nonlinear Kato classes.