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.     

Best quadrature formula on Sobolev class with Chebyshev weight

Using best interpolation function based on a given function information, we present a best quadrature rule of function on Sobolev class KW^r[-1,1]${\mathit{KW}}^r[-1,1]$ with Chebyshev weight. The given function information means that the values of a function f∈KW^r[-1,1]$f\in {\mathit{KW}}^r[-1,1]$ and its derivatives up to r-1$r-1$ order at a set of nodes x$\mathbf{x}$ are given. Error bounds are obtained, and the method is illustrated by some examples.     

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

A quasiplane f(V)$f(V)$ is the image of an n-dimensional Euclidean subspace V   of R^N${\mathbb{R}}^N$ (1≤n≤N−1$1\le n\le N-1$) under a quasiconformal map f:R^N→R^N$f:{\mathbb{R}}^N\to {\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of Rⁿ${\mathbb{R}}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−n$N-n$. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.     

On the “viscous incompressible fluid + rigid body” system with Navier conditions

In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole system “viscous incompressible fluid + rigid body” is assumed to occupy the full space R³${\mathbb{R}}^3$. We start by proving the existence of global weak solutions to the Cauchy problem. Then, we exhibit several properties of these solutions. First, we show that the added-mass effect can be computed which yields better-than-expected regularity (in time) of the solid velocity-field. More precisely we prove that the solid translation and rotation velocities are in the Sobolev space H¹$H^1$. Second, we show that the case with the body fixed can be thought as the limit of infinite inertia of this system, that is when the solid density is multiplied by a factor converging to +∞. Finally we prove the convergence in the energy space of weak solutions “à la Leray” to smooth solutions of the system “inviscid incompressible fluid + rigid body” as the viscosity goes to zero, till the lifetime T   of the smooth solution of the inviscid system. Moreover we show that the rate of convergence is optimal with respect to the viscosity and that the solid translation and rotation velocities converge in H¹(0,T)$H^1(0,T)$.     

Spectral element methods for parabolic problems

A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k   is proportional to h²$h^2$. At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.     

Remarks on a lemma by Jacques-Louis Lions

Let Ω   be a bounded and connected open subset of R^N${\mathbb{R}}^N$ with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω  . It is shown in this Note that a fundamental characterization of the space L²(Ω)$L^2(\Omega )$ due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.     

A characterization of quintic helices

A polynomial curve of degree 5, α,$\alpha ,$ is a helix if and only if both ∥α^′∥$\parallel {\alpha }^{\prime }\parallel$ and ∥α^′∧α^″∥$\parallel {\alpha }^{\prime }\wedge {\alpha }^{″}\parallel$ are polynomial functions.     

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let Ω   be a smooth bounded simply connected domain in R²${\mathbb{R}}^2$. We investigate the existence of critical points of the energy _Eε(u)=1/2_∫Ω|∇u|²+1/(4ε²)_∫Ω(1−|u|²)²$E_{\epsilon }(u)=1/2{\int }_{\Omega }{|\mathrm{\nabla }u|}^2+1/(4{\epsilon }^2){\int }_{\Omega }{(1-{|u|}^2)}^2$, where the complex map u has modulus one and prescribed degree d on the boundary. Under suitable nondegeneracy assumptions on Ω, we prove existence of critical points for small ε. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disk. Next, we prove that critical points exist in “most” of the domains.     

Filtrations induced by continuous functions

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to Rⁿ${\mathbb{R}}^n$. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable 1-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multi-dimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed.     

Domain of validity of Szegö quadrature formulas

As is well known, the n  -point Szegö quadrature formula integrates correctly any Laurent polynomial in the subspace span{1/z^n-1,…,1/z,1,z,…,z^n-1}$1/z^{n-1},\dots ,1/z,1,z,\dots ,z^{n-1}\}$. In this paper we enlarge this subspace. We prove that a set of 2n$2n$ linearly independent Laurent polynomials are integrated correctly. The obtained result is used for the construction of Szegö quadrature formulas. Illustrative examples are given.     

Random triangle removal

Starting from a complete graph on n   vertices, repeatedly delete the edges of a uniformly chosen triangle. This stochastic process terminates once it arrives at a triangle-free graph, and the fundamental question is to estimate the final number of edges (equivalently, the time it takes the process to finish, or how many edge-disjoint triangles are packed via the random greedy algorithm). Bollobás and Erd?s (1990) conjectured that the expected final number of edges has order n^3/2$n^{3/2}$. An upper bound of o(n²)$o(n^2)$ was shown by Spencer (1995) and independently by Rödl and Thoma (1996). Several bounds were given for variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald (1999)), while the best known upper bound for the original question of Bollobás and Erd?s was n^7/4+o(1)$n^{7/4+o(1)}$ due to Grable (1997). No nontrivial lower bound was available.     

Age-time continuous Galerkin methods for a model of population dynamics

Continuous Galerkin finite element methods in the age-time domain are proposed to approximate the solution to the model of population dynamics with unbounded mortality (coefficient) function. Stability of the method is established and a priori L²$L^2$-error estimates are obtained. Treatment of the nonlocal boundary condition is straightforward in this framework. The approximate solution is computed strip by strip marching in time. Some numerical examples are presented.     

Quasiconvexification in 3-D for a variational reformulation of an optimal design problem in conductivity

We analyze a typical 3-D conductivity problem which consists in seeking the optimal layout of two materials in a given design domain Ω⊂R³$\Omega \subset {\mathbb{R}}^3$ by minimizing the L²$L^2$-norm of the electric field under a constraint on the amount on each material that we can use. We utilize a characterization of the 3-D divergence-free vector fields which is especially appropriate for a variational reformulation. By using gradient Young measures as a main tool, we can give an explicit form of the “constrained quasiconvexification” of the cost density. This result is similar to the one in the 2-D situation. However, the characterization of the divergence-free vector fields introduces a certain nonlinearity in the problem that needs to be addressed properly.     

Light-likeness of boundaries of extremal surfaces of mixed type in physical space–times

In this paper, we prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space–time R¹⁺ⁿ$R^{1+n}$(n>1)$(n>1)$, in particular we improve Gu's theorem on the light-likeness of boundaries of extremal surfaces in R¹⁺²$R^{1+2}$ and prove the light-likeness of boundaries of smooth extremal surfaces of mixed type in general physical space–times. As a consequence, we show that a curve moving in a physical space–time keeps its like-property and the boundary only exists when its world sheet at the initial time has light-like points. This implies that any extremal surface of mixed type is generated by an “initial curve of mixed type”.