A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness

In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^{d} \subset\mathbb{R}^{d+1}$ . The reproducing kernel is given by an integral representation using the truncated power function $(\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}$ supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β=1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003). We show that the reproducing kernel is a sum of the Euclidean distance ∥x?y∥ of the arguments of the kernel raised to the power of 2β?1 and an adjustment in the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β?1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For $\beta\in\mathbb{N}$ , the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Stolarsky’s invariance principle states that the sum of all mutual distances among N points plus a certain multiple of the spherical cap $\mathbb{L}_{2}$ -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap $\mathbb{L}_{2}$ -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over $\mathbb{S}^{d}$ of smoothness (d+1)/2 provided with the reproducing kernel 1?C _d ∥x?y∥ for some constant C _d . Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap $\mathbb{L}_{2}$ -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over $\mathbb{S}^{d}$ of arbitrary smoothness s=β?1/2+d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0,1] ^s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).     

Closed form of the Fourier–Borel Kernel in the Framework of Clifford Analysis

In this paper we derive for the even dimensional case a closed form of the Fourier–Borel kernel in the Clifford analysis setting. This kernel is obtained as the monogenic component in the Fischer decomposition of the exponential function ${e^{\langle \underline{x}, \underline{u} \rangle}}$ where ${\langle . , . \rangle}$ denotes the standard inner product on the m-dimensional Euclidean space. A first approach based on Clifford analysis techniques leads to a conceptual formula containing the Gamma operator and the so-called Clifford–Bessel function, two fundamental objects in the theory of Clifford analysis. To obtain an explicit expression for the Fourier–Borel kernel in terms of a finite sum of Bessel functions, this formula remains however hard to work with. To that end we have also elaborated a more direct approach based on special functions leading to recurrence formulas for a closed form of the Fourier–Borel kernel.     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

On Some Noteworthy Pairs of Ideals in Mod-R

An additive functor $F \colon {\mathcal A}\to{\mathcal B}$ between preadditive categories $\mathcal A$ and $\mathcal B$ is said to be a local functor if, for every morphism $f\colon A\to A'$ in $\mathcal A$ , F(f) isomorphism in $\mathcal B$ implies f isomorphism in $\mathcal A$ . We show that there exist several pairs $(\mathcal I_1,\mathcal I_2)$ of ideals of $\mathcal A$ for which the canonical functor $\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2$ is a local functor. In most of our examples, the category $\mathcal A$ is a full subcategory of the category Mod?-R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod?-R into its spectral category Spec(Mod?-R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ⁽¹⁾ and Σ₍₁₎ of the first derived functors P ⁽¹⁾ and F ₍₁₎ of P and F, respectively; (4) the kernels of suitable functors Hom and ? and their first derived functors ${\rm Ext}^1_R$ and ${\rm Tor}^R_1$ .     

Topological degree for solutions of fourth order mean field equations

We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C ^2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

Cohomology and Support Varieties for Restricted Lie Superalgebras

Let $(\mathfrak{g}, [p]) $ be a restricted Lie superalgebra over an algebraically closed field k of characteristic p?>?2. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$ . In this paper we prove that the cohomology ring $\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ -supermodules. We also show that support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ - supermodules satisfy the desirable properties of a support variety theory.     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time.     

Dynamical system with boundary control associated with a symmetric semibounded operator

Let L ₀ be a closed densely defined symmetric semibounded operator with nonzero defect indices in a separable Hilbert space $\mathcal H$ . It determines a Green system $\{{\mathcal H}, {\mathcal B}; L_0, \Gamma_1, \Gamma_2\}$ , where ${\mathcal B}$ is a Hilbert space, and the $\Gamma_i: {\mathcal H} \to \mathcal B$ are operators connected by the Green formula $$ (L_0^*u, v)_{\mathcal H}-(u,L_0^*v)_{\mathcal H} =(\Gamma_1 u, \Gamma_2 v)_{\mathcal B} - (\Gamma_2 u, \Gamma_1 v)_{\mathcal B}. $$ The boundary space $\mathcal B$ and the boundary operators Γ _i are chosen canonically in the framework of the Vishik theory. With the Green system one associates a dynamical system with boundary control (DSBC): $$ \begin{array}{lll} && u_{tt}+L_0^*u = 0, \quad u(t) \in {\mathcal H}, \quad t>0,\\ && u\big|_{t=0}=u_t\big|_{t=0}=0, \\ && \Gamma_1 u = f, \quad f(t) \in {\mathcal B},\quad t \geq 0. \end{array} $$ We show that this system is controllable if and only if the operator L ₀ is completely non-self-adjoint. A version of the notion of wave spectrum of L ₀ is introduced. It is a topological space determined by L ₀ and constructed from reachable sets of the DSBC. Bibliography: 15 titles.     

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on L _p (?₊) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L ₂(?₊) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.     

Nonlinear superharmonic functions and supersolutions

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations ${-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}$ and their generalizations ${-\nabla \cdot A(x,\nabla u) = \mu}$ ,where ${A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}$ , the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpeläinen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation ${-\nabla \cdot A(x,\nabla u) = f(x)}$ .     

Global analytic hypoellipticity of the\bar \partial-Neumann problem on circular domains

In this paper we show that if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded pseudoconvex circular domain with real analytic defining functionr(z) such that \(\sum\limits_{k = 1}^n {z_k \frac{{\partial r}}{{\partial z_k }}} \ne 0\) for allz near the boundary, then the solutionu to the \(\bar \partial\) -Neumann problem, $$square u = (\bar \partial \bar \partial * + \bar \partial *\bar \partial )u = f,$$ is real analytic up to the boundary, if the given formf is real analytic up to the boundary. In particular, if \(D \subseteq \mathbb{C}^n ,n \geqq 2\) , is a smooth bounded complete Reinhardt pseudoconvex domain with real analytic boundary. Then ? is analytic hypoelliptic.     

Singularly perturbed nonlinear Dirichlet problems involving critical growth

We consider the following singularly perturbed nonlinear elliptic problem: $$\begin{array}{ll}-\varepsilon^{2}\Delta u + u=f(u),\; u > 0\, {\rm on}\, \Omega,\; u = 0\, {\rm on}\, \partial \Omega,\end{array}$$ where Ω is a bounded domain in ${\mathbb{R}^N (N \ge 3)}$ with a boundary ${\partial \Omega \in C^2}$ and the nonlinearity f is of critical growth. In this paper, we construct a solution ${u_\varepsilon}$ of the above problem which exhibits one spike near a maximum point of the distance function from the boundary ?Ω under a critical growth condition on f. Our result complements the study made in [9] in the sense that, in that paper, only the subcritical growth was considered.     

Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C ² boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω.     

A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

(A) The celebrated Gaussian quadrature formula on finite intervals tells us that the Gauss nodes are the zeros of the unique solution of an extremal problem. We announce recent results of Damelin, Grabner, Levesley, Ragozin and Sun which derive quadrature estimates on compact, homogenous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. (B) Given $\mathcal{X}$ , some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, $\mathcal{P} \subset \mathcal{X}$ , with a small energy or discrepancy. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that these two measures of quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^2(=\mathcal{X}\times\mathcal{X}) \to \mathbb{R}$ . The error of approximating the integral $\int_{\mathcal{X}} f(\mathbf{\mathit{x}}) \, {\rm d} \mu(\mathbf{\mathit{x}})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms the energy or discrepancy of $\mathcal{P}$ . The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. (C) Let $\mathcal{X}$ be the orbit of a compact, possibly non Abelian group, $\mathcal{G}$ , acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$ . This allows us to calculate explicit representations of equilibrium measures. There is an extensive literature on the topics (A–C). We emphasize that this paper surveys recent work of Damelin, Grabner, Levesley, Hickernell, Ragozin, Sun and Zeng and does not mean to serve as a comprehensive survey of all recent work covered by the topics (A–C).     

The Robin and Wentzell-Robin Laplacians on Lipschitz Domains

Let $\Omega\subset{\Bbb R}^N$ be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ generates a holomorphic $C_0$ -semigroup of angle $\pi/2$ on $C(\overline{\Omega})$ if $0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)$ . With the same assumption on $\Omega$ and assuming that $0\le\beta\in L^{\infty}(\partial \Omega)$ , we show in the second part that one can define a realization of the Laplacian on $C(\overline{\Omega})$ with Wentzell-Robin boundary conditions $\Delta u+\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ and this operator generates a $C_0$ -semigroup.     

A maximum modulus theorem for the Oseen problem

Classical solutions of the Oseen problem are studied on an exterior domain Ω with Ljapunov boundary in R ³. It is proved a maximum modulus estimate of the following form: If ${{\bf u}\in C^2(\Omega)^3\cap C^0(\overline \Omega)^3}$ and ${p \in C^1(\Omega ), -\Delta {\bf u}+2\lambda \partial_1 {\bf u}+\nabla p=0, \nabla \cdot {\bf u}=0}$ in Ω, and if ${|{\bf u}| \le M}$ on ${\partial \Omega , \limsup |{\bf u}({\bf x})|\le M}$ as ${|{\bf x}|\to \infty }$ , then ${|{\bf u}({\bf x})|\le c M}$ in Ω. Here the constant c depends only on Ω and λ.