Removable Singularities Research Articles

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.

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace polynomial matrix, and $I$ is the unit matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of the essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires finding of the $\rho$ zeros of the Baker-Akhiezer function ($\rho$ is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-$\rho$ polynomial and solution of a certain linear algebraic system of $\rho$ equations.

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

The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a null Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved null Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve Holder regularity and stability theorems of [19].

An analogy strategy for transformation optics

We introduce an analogy strategy to design transformation optical devices. Based on the similarities between field lines in different physical systems, the trajectories of light can be intuitively determined to curve in a gentle manner, and the resulting materials are isotropic and nonmagnetic. Furthermore, the physical meaning of the analogue problems plays a key role in the removal of dielectric singularities. We illustrate this approach by creating two designs of carpet cloak and a collimating lens as representative examples in two- and three-dimensional spaces, respectively. The analogy strategy not only reveals the intimate connections between different physical disciplines, such as optics, fluid mechanics and electrostatics, but also provides a heuristic pathway to designing advanced photonic systems.

Removable Singularities for Anisotropic Elliptic Equations

We study a class of quasi-linear elliptic equations with model representative \(\sum _{i=1}^{n}(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}})_{x_{i}}=0\), which solutions have singularities on a smooth manifold. We establish the condition for removability of singularity on a manifold for solutions of such equations.

Removable singularities and singular solutions of semilinear elliptic equations

This note proves that a closed set with appropriate properties is removable for solutions of semilinear elliptic equations satisfying a certain growth condition near that set. Also, we give the existence theorem of positive solutions with singularities on a prescribed compact set, which shows that a growth condition in our removability theorem is optimal.

Some properties of weak solutions to the weighted p-Laplacian near an isolated singular point

We study the weighted p-Laplacian with an isolated singular point, and discuss some properties of solutions near such a point, including removable singularities, boundedness and growth rate estimates. In particular, some relevant critical exponents are exactly determined.

Gluon propagator in fractional analytic perturbation theory

We consider the gluon propagator in the Landau gauge at low spacelike momenta and with the dressing function $Z({Q}^{2})$ at the two-loop order. We incorporate the nonperturbative effects by making the (noninteger) powers of the QCD coupling in the dressing function $Z({Q}^{2})$ analytic (holomorphic) via the fractional analytic perturbation theory (FAPT) model, and simultaneously introducing the gluon dynamical mass in the propagator as motivated by the previous analyses of the Dyson-Schwinger equations. The obtained propagator has behavior compatible with the unquenched lattice data (${N}_{f}=2+1$) at low spacelike momenta $0.4\text{ }\text{ }\mathrm{GeV}<Q\ensuremath{\lesssim}10\text{ }\text{ }\mathrm{GeV}$. We conclude that the removal of the unphysical Landau singularities of the powers of the coupling via the (F)APT prescription, in conjunction with the introduction of the dynamical mass $M\ensuremath{\approx}0.62\text{ }\text{ }\mathrm{GeV}$ of the gluon, leads to an acceptable behavior of the propagator in the infrared regime.

Removal of singularities and Gromov compactness for symplectic vortices

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compact- ication by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vor- tices that allows us to apply techniques used by McDu and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.

Removable singularities for nonlinear subequations

Removable singularities for nonlinear subequations

Automatic computer procedure for generating exact and analytical kinetic energy operators based on the polyspherical approach: General formulation and removal of singularities

We present new techniques for an automatic computation of the kinetic energy operator in analytical form. These techniques are based on the use of the polyspherical approach and are extended to take into account Cartesian coordinates as well. An automatic procedure is developed where analytical expressions are obtained by symbolic calculations. This procedure is a full generalization of the one presented in Ndong et al., [J. Chem. Phys. 136, 034107 (2012)]. The correctness of the new implementation is analyzed by comparison with results obtained from the TNUM program. We give several illustrations that could be useful for users of the code. In particular, we discuss some cyclic compounds which are important in photochemistry. Among others, we show that choosing a well-adapted parameterization and decomposition into subsystems can allow one to avoid singularities in the kinetic energy operator. We also discuss a relation between polyspherical and Z-matrix coordinates: this comparison could be helpful for building an interface between the new code and a quantum chemistry package.

A-Harmonic operator in the Dirac system

In this paper, we show how an A-harmonic operator arises from A-Dirac systems under natural growth condition. By the method of removable singularities for solutions to A-Dirac systems with natural growth conditions, we establish the fact that A-harmonic equations are the real part of the corresponding A-Dirac systems.

Removable singularities of semilinear parabolic equations

This paper extends the recent result due to Hsu (2010) about removable singularities of semilinear parabolic equations. Our result is applicable to solutions of equations of the form ∆u + @tu = juj p 1 u with 0 p < n=(n 2). The proof is based on the parabolic potential theory and an iteration argument. Also, we prove that if 0 < p < (n + 2)=n, then integral solutions of semilinear parabolic equations with nonlinearity depending on space and time variables and u p are locally bounded. This implies that the blow-up for continuous solutions is complete.

The relation between A-harmonic operator and A-Dirac system

In this paper, we show how an A-harmonic operator arises from Dirac systems under controllable growth condition. By the method of removable singularities for solutions to the A-Dirac system with controllable growth conditions, we establish the fact that an A-harmonic operator is a real part of the corresponding A-Dirac systems.

Riesz capacity, maximum principle and removable sets of fully nonlinear second order elliptic operators

In this paper we show sufficient conditions for the extended maximum principle and the removable singularities for viscosity solutions of fully nonlinear second-order elliptic equations via Riesz and logarithmic capacity.

Quasilinear equations with source terms on Carnot groups

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane–Emden type with measure data on a Carnot group G of arbitrary step. The quasilinear part involves operators of the p-Laplacian type ∆G, p , 1 < p < ∞. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff’s type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub-Laplacian (p = 2) on the Heisenberg group.

On the boundary behaviour of solutions to the Beltrami equations

It is shown that each homeomorphic solution to the Beltrami equation is the so-called lower Q-homeomorphism with Q(z) = K μ(z) where K μ(z) is the dilatation quotient of this equation. The theory of the boundary behaviour and the removability of singularities of such solutions are developed on this basis.

Surfaces of constant curvature in [formula omitted] with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature K>0 in R3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in S2 with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R3, i.e. compact convex surfaces of constant curvature K>0 with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

On the existence of best Mitscherlich, Verhulst, and West growth curves for generalized least-squares regression

Practical difficulties arise in fitting Mitscherlich, Verhulst, or West growth curves to data. Obstacles include divergent iterations, negative values for theoretically positive parameters, or the absence of any best-fitting curve. An analysis reveals that such obstacles occur near removable singularities of the objective function to be minimized for the regression. Such singularities lie at the transition to different types of curves, including exponentials, hyperbolae, lines, and step functions. Removing the singularities fits all such curves into a connected compactified topological space, which guarantees the existence of a global minimum for the continuous objective function, and which also provides a smooth and transparent transition from one type of curve to another.

Some Properties of Solutions to Weakly Hypoelliptic Equations

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.