The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients $a_{l}^{m}$ of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form ?? ^p or |????? ₀| ^p respectively, where ?? is the co-latitude and p>?1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, $E(l) = \sqrt{\sum_{m} (a_{l}^{m})^{2}}$ where l and m are the spherical harmonic degree and order, of l ^?(p+3/2) or l ^?(p+1) respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of E(l) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface.     

Nonuniqueness of the traveling wave speed for harmonic heat flow

Given any wave speed c∈R, we construct a traveling wave solution of ut=Δu+²|∇u|u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x₃=±∞. Here u is a director field with values in S²⊂R³: |u|=1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.     

The Hele-Shaw problem with surface tension in a half-plane

In this paper, we consider the Hele-Shaw problem in a 2-dimensional fluid domain Ω(t) which is constrained to a half-plane. The boundary of Ω(t) consist of two components: Γ₀(t) which lies on the boundary of the half-plane, and Γ(t) which lies inside the half-plane. On Γ(t) we impose the classical boundary conditions with surface tension, and on Γ₀(t) we prescribe the normal derivative of the fluid pressure. At the point where Γ₀(t) and Γ(t) meet, there is an abrupt change in the boundary condition giving rise to a singularity in the fluid pressure. We prove that the problem has a unique solution with smooth free boundary Γ(t) for some small time interval.     

A finite element method scheme for boundary value problems with noncoordinated degeneration of input data

A finite element method scheme is constructed for boundary value problems with noncoordinated degeneration of input data and singularity of a solution. We look at a rate with which an approximate solution by the proposed finite element method converges toward an exact R _ν -generalized solution in the weight set W _2,ν*+β ^2+1/1 (Ω, δ), and establish estimates for the finite element approximation.     

Superminimal surfaces in the 6-sphere

In this article, we use the harmonic sequence associated to a weakly conformal harmonic map f: S → S ⁶ in order to determine explicit examples of linearly full almost complex 2-spheres of S ⁶ with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry and this allows us to determine the moduli space of such curves with suitably small area. We also characterize projectively equivalent almost complex curves of S ⁶ in terms of G ₂ ^? -equivalence of their directrix curves.     

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ? that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x ₀. For small values of ?, a boundary layer with the typical width of ? appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x ₀, 0), a transient (moving in time) layer with the typical width of ?^1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ?-uniformly on the entire set $\bar G$ , approximate the diffusion flow (i.e., the product ?(?/?x)u(x, t)) on the set $\bar G^ * = \bar G\backslash \{ (x_0 ,0)\} $ , and approximate the derivative (?/?x)u(x, t) on the same set outside the m-neighborhood of the boundary layer. The approximation of the derivatives ?²(?²/?x ²)u(x, t) and (?/?t)u(x, t) on the set $\bar G^ * $ is also examined.     

Discrete invariant imbedding for singular boundary value problems with a regular singularity

We describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval [0,b] for a linear second order ordinary differential equation with a singularity of the first kind at x = 0. By employing the series expansion on (0, δ], where δ is near the singularity, we first replace it by a regular problem on some interval [δ, b]. The discrete invariant imbedding method is then described to solve the problem over the reduced interval. The stability analysis of the method is discussed. Some model problems are solved, and the numerical results compared with those of other methods.     

Boundary singularities of solutions of N-harmonic equations with absorption

We study the boundary behaviour of solutions u of −ΔNu+|u|^q−1u=0 in a bounded smooth domain Ω⊂RN subject to the boundary condition u=0 except at one point, in the range q>N−1. We prove that if q?2N−1 such an u is identically zero, while, if N−1<q<2N−1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.     

Trigonal Gorenstein curves

We notice that the Maroni invariant of a trigonal Gorenstein curve of arithmetic genus g larger than four may be equal to zero, and we show that this happens if and only if the g₃¹ admits a non-removable base point, which is necessarily a singularity of the curve. We realize and study trigonal curves on rational scrolls, which in the case, where the g₃¹ admits a base point Q, degenerate to a cone with vertex Q.     

Topologie locale des méthodes de Newton cubiques: plan paramétrique

Cubic Newton 's methods are rational maps having three distinct super-attracting fixed points and a single free critical point. They form, up to conjugation, a family N_λ parametrized by Λ = ℂ\{0,±3/2}, and we denote by ℋ₀ the set of λ for which the free critical point of N_λ is in the immediate basin of one of the super-attracting fixed points. In this Note, we show that the boundary of each connected component of ℋ₀ is a Jordan curve. For this, we determine in Λ regions on which the dynamics of N_λ can be described by a fixed combinatorial model.     

Oscillations of a gas in an open-ended tube near resonance

The pressure as function of time was measured near resonance in different axial locations of an open-ended tube. Flow visualisation showed that transition to turbulence was not influenced by the strong disturbance of the open end, except in a region near the open end which had a length of about three particle displacements. The pressure readings were decomposed into the first, second and third harmonic and compared with two different theories. In one case, the linearized theory for the oscillating flow in a tube was fitted to the boundary conditions, the obvious one at the piston and a model at the open end. In the second case, the nonlinear theory of Chester [1] was used. Both theories assume a relation between pressure and velocity at the open end that contains two free constants. The constants were determined by comparing the amplitude of the first and the second harmonic ofone pressure measurement with the theoretical predictions. Once the constants are fixed, the pressurep(ωt, x/L) is completely determined. For weak nonlinear effects, the pressure is essentially determined by one constantα(=k ₂) and the second constantβ(=k ₁) loses its significance. For the range of parameters given there isα=0.825±0.015. A very good approximation of the pressure near resonance can therefore be calculated with the following simple boundary condition at the open end $$p_E = \frac{{4\alpha }}{{3\pi }}\rho \hat u_E u_E = 0.350 \rho \hat u_E u_E .$$ Both theories predict a resonance frequency slightly above the experimental one. Changing Levine and Schwingers [2], end correction from 0.6133R to 1R eliminates the discrepancy for all tube lengths. For the first harmonic the variation of the amplitude and the phase of the pressure signal withω andx is very well predicted by both theories. The nonlinear theory describes also the small second and third harmonics fairly well while the linear theory predicts only the correct order of magnitude of these higher harmonics. The constantα that determines the energy loss at the open end shows an apparent increase if the boundary layer on the tube wall becomes turbulent. This occurs for \(A = 2\hat u/\sqrt {v\omega } \geqq 550\) to 750 which is close to the value observed in a tube with a closed end.     

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ?². Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.     

Existence of weak positive solutions to a nonlinear PDE system around a triple phase boundary, coupling domain and boundary variables

We consider a 2D nonlinear system of PDEs representing a simplified model of processes near a triple-phase boundary (TPB) in cathode catalyst layer of hydrogen fuel cells. The particularity of this system is the coupling of a variable satisfying a PDE in the interior of the domain with another variable satisfying a differential equation (DE) defined only on the boundary, through an adsorption-desorption equilibrium mechanism. The system includes also an isolated singular boundary condition which models the flux continuity at the contact of the TPB with a subdomain. By freezing certain terms we transform the nonlinear PDE system to an equation, which has a variational formulation. We prove several L^∞ and W1,p a priori estimates and then by using Schauder fixed point theorem we prove the existence of a weak positive bounded solution.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

Characterization of solutions to the generalized Cauchy-Riemann system

For a generalized biaxially symmetric potential U on a semi-disk D⁺, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ?D⁺. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ?D⁺.     

Asymptotic Boundary Value Problem of Harmonic Maps via Harmonic Boundary

We prove that given any continuous data f on the harmonic boundary of a complete Riemannian manifold with image within a ball in the normal range, there exists a harmonic map from the manifold into the ball taking the same boundary value at each harmonic boundary point as that of f.     

Spatial curve singularities and the monster/semple tower

The Monster tower ([MZ01], [MZ10]), known as the Semple Tower in Algebraic Geometry ([Sem54], [Ber10]), is a tower of fibrations canonically constructed over an initial smooth n-dimensional base manifold. Each consecutive fiber is a projective n — 1 space. Each level of the tower is endowed with a rank n distribution, that is, a subbundle of its tangent bundle. The pseudogroup of diffeomorphisms of the base acts on each level so as to preserve the fibration and the distribution. The main problem is to classify orbits (equivalence classes) relative to this action. Analytic curves in the base can be prolonged (= Nash blown-up) to curves in the tower which are integral for the distribution. Prolongation yields a dictionary between singularity classes of curves in the base n-space and orbits in the tower. This dictionary yielded a rather complete solution to the classification problem for n = 2 ([MZ10]). A key part of this solution was the construction of the ‘RVT’ classes, a discrete set of equivalence classes built from verifying conditions of transversality or tangency to the fiber at each level ([MZ10]). Here we define analogous ‘RC’ classes for n > 2 indexed by words in the two letters, R (for regular, or transverse) and C (for critical, or tangent). There are 2 ^k?1 such classes of length k and they exhaust the tower at level k. The codimension of such a class is the number of C’s in its word. We attack the classification problem by codimension, rather than level. The codimension 0 class is open and dense and its structure is well known. We prove that any point of any codimension 1 class is realized by a curve having a classical A _2k singularity (k depending on the type of class). Following ([MZ10]) we define what it means for a singularity class in the tower to be “tower simple”. The codimension 0 and 1 classes are tower simple, and tower simple implies simple in the usual sense of singularity. Our main result is a classification of the codimension 2 tower simple classes in any dimension n. A key step in the classification asserts that any point of any codimension 2 singularity is realized by a curve of multiplicity 3 or 4. A central tool used in the classification are the listings of curve singularities due to Arnol’d ([Arn99], Bruce-Gaffney ([BG82]), and Gibson-Hobbs ([GH93]). We also classify the first occurring truly spatial singularities as subclasses of the codimension 2 classes. (A point or a singularity class is “spatial” if there is no curve which realizes it and which can be made to lie in some smooth surface.) As a step in the classification theorem we establish the existence of a canonical arrangement of hyperplanes at each point, lying in the distribution n-plane at that point. This arrangement leads to a coding scheme finer than the RC coding. Using the arrangement coding we establish the lower bound of 29 for the number of distinct orbits in the case n = 3 and level 4. Finally, Mormul ([Mor04], [Mor09]) has defined a different coding scheme for singularity classes in the tower and in an appendix we establish some relations between our coding and his.     

On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W _p ¹ on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.