Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality _Δφu?f(u)?(|_∇0u|)${\mathrm{\Delta }}_{\phi }u?f(u)?(|{\mathrm{\nabla }}_{0}u|)$ on the Heisenberg group under a generalized Keller–Osserman condition. The operator _Δφu${\mathrm{\Delta }}_{\phi }u$ is the φ  -Laplacian defined by _div0(|_∇0u|⁻¹φ(|_∇0u|)_∇0u)${\mathrm{div}}_0({|{\mathrm{\nabla }}_{0}u|}^{-1}\phi (|{\mathrm{\nabla }}_{0}u|){\mathrm{\nabla }}_{0}u)$ and φ, f and ? satisfy mild structural conditions. In particular, ? is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Characterization of an extremal sum of ridge functions

The approximation problem considered in the paper is to approximate a continuous multivariate function f(x)=f(_x1,…,_xd)$f(\mathbf{x})=f(x_1,\dots ,x_d)$ by sums of two ridge functions in the uniform norm. We give a necessary and sufficient condition for a sum of two ridge functions to be a best approximation to f(x)$f(\mathbf{x})$. This main result is next used in a special case to obtain an explicit formula for the approximation error and to construct one best approximation. The problem of well approximation by such sums is also considered.     

Regularity properties of the solutions to the 3D Maxwell–Landau–Lifshitz system in weighted Sobolev spaces

This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of _∂tm,_∂tE${\mathrm{\partial }}_t\mathbf{m},{\mathrm{\partial }}_t\mathbf{E}$ and _∂tH${\mathrm{\partial }}_t\mathbf{H}$ in the spaces L^∞(I,L²(Ω))$L^{\infty }(I,L^2(\Omega ))$ and L²(I,W^1,2(Ω))$L^2(I,W^{1,2}(\Omega ))$ we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme.     

Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

−Δu−λu=a₊(x)|u|^q−1u−a₋(x)|u|^p−1u+h(x,u)$-\mathrm{\Delta }u-\lambda u=a_{+}(x){|u|}^{q-1}u-a_{-}(x){|u|}^{p-1}u+h(x,u)$

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)_utt−div(A(x)∇u)+a(x)_ut=0$c(x)u_{tt}-\mathrm{div}(A(x)\mathrm{\nabla }u)+a(x)u_t=0$ in Rⁿ${\mathbb{R}}^n$ are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L²$L^2$ estimates for spatial derivatives.     

A core-free semicovering of the Hawaiian Earring

The connected covering spaces of a connected and locally path-connected topological space X   can be classified by the conjugacy classes of those subgroups of _π1(X,x)${\pi }_1(X,x)$ which contain an open normal subgroup of _π1(X,x)${\pi }_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring H$\mathbb{H}$ with corresponding open subgroup of _π1(H)${\pi }_1(\mathbb{H})$ which does not contain any   nontrivial normal subgroup of _π1(H)${\pi }_1(\mathbb{H})$.     

Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality |(i_∂t+Δ)u|?|V(x)u|$|(i{\partial }_t+\mathrm{\Delta })u|?|V(x)u|$ in Rⁿ⁺¹${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials V(x)$V(x)$ in Rⁿ${\mathbb{R}}^n$. Our method is based on a new type of Carleman estimates for the operator i_∂t+Δ$i{\partial }_t+\mathrm{\Delta }$ on Rⁿ⁺¹${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

The Schwarz genus of the Stiefel manifold

In this paper we compute: the Schwarz genus of the Stiefel manifold _Vk(Rⁿ)$V_k({\mathbb{R}}^n)$ with respect to the action of the Weyl group _Wk:=(Z/2)^k?_Sk$W_k:={(\mathbb{Z}/2)}^k?{\mathfrak{S}}_k$, and the Lusternik–Schnirelmann category of the quotient space _Vk(Rⁿ)/_Wk$V_k({\mathbb{R}}^n)/W_k$. Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space.