Recursos de colección

1. Stable Higgs $G$-sheaves

Gómez , Tomás L.; Sols , Ignacio
For a connected reductive group $G$, we generalize the notion of (semi)stable Higgs $G$-bundles on curves to smooth projective schemes of higher dimension, allowing also Higgs $G$-sheaves, and construct the corresponding moduli space.

2. On the NLS dynamics for infinite energy vortex configurations on the plane

Bethuel , Fabrice; Jerrard , Robert L.; Smets , Didier
We derive the asymptotical dynamical law for Ginzburg-Landau vortices in the plane under the Schrödinger dynamics, as the Ginz\-burg-Landau parameter goes to zero. The limiting law is the well-known point-vortex system. This result extends to the whole plane previous results of [Colliander, J.E. and Jerrard, R.L.: Vortex dynamics for the Ginzburg-Landau-Schrödinger equation. Internat. Math. Res. Notices 1998, no. 7, 333-358; Lin, F.-H. and Xin, J.\,X.: On the incompressible fluid limit and the vortex motion law of the nonlinear Schr\"{o}dinger equation. Comm. Math. Phys. 200 (1999), 249-274] established for bounded domains, and holds for arbitrary degree at infinity. When this degree is non-zero, the total Ginzburg-Landau energy is infinite.

3. Measure density and extendability of Sobolev functions

Hajłasz , Piotr; Koskela , Pekka; Tuominen , Heli
We study necessary and sufficient conditions for a domain to be a Sobolev extension domain in the setting of metric measure spaces. In particular, we prove that extension domains must satisfy a measure density condition.

4. Notes on the roots of Steiner polynomials

Henk , Martin; Hernández Cifre , María A.
We study the location and the size of the roots of Steiner polynomials of convex bodies in the Minkowski relative geometry. Based on a problem of Teissier on the intersection numbers of Cartier divisors of compact algebraic varieties it was conjectured that these roots have certain geometric properties related to the in- and circumradius of the convex body. We show that the roots of 1-tangential bodies fulfill the conjecture, but we also present convex bodies violating each of the conjectured properties.

5. On the verbal width of finitely generated pro-$p$ groups

Jaikin-Zapirain , Andrei
Let $p$ be a prime. It is proved that a non-trivial word $w$ from a free group $F$ has finite width in every finitely generated pro-$p$ group if and only if $w\not \in (F^\prime)^{p} F^{\prime\prime}$. Also it is shown that any word $w$ has finite width in a compact $p$-adic group.

6. Quasilinear equations with natural growth

Arcoya , David; Martínez-Aparicio , Pedro J.
We study the existence of positive solution $w\in H_0^1(\Omega)$ of the quasilinear equation $-\Delta w+ g(w)|\nabla w|^2=a(x)$, $x\in \Omega$, where $\Omega$ is a bounded domain in $\mathbb R^N$, $0\leq a\in L^\infty (\Omega )$ and $g$ is a nonnegative continuous function on $(0,+\infty)$ which may have a singularity at zero.

7. Non-uniqueness in a free boundary problem

Bennewitz , Björn
We show that a result of Lewis and Vogel on uniqueness in a free boundary problem for the $p$-Laplace operator is sharp in two dimensions.

8. Geometric optics with critical vanishing viscosity for one-dimensional semilinear initial value problems

Junca , Stéphane
We study the propagation of high frequency oscillations for one dimensional semi-linear hyperbolic systems with small parabolic perturbations. We obtain a new degenerate parabolic system for the profile, and valid an asymptotic development in the spirit of Joly, Métivier and Rauch.

9. $L^2$ boundedness for commutator of rough singular integral with variable kernel

Chen , Yanping; Ding , Yong
In this paper the authors prove the $L^2(\mathbb{R}^n)$ boundedness of the commutator of the singular integral operator with rough variable kernels, which is a substantial improvement and extension of some known results.

10. Interpolation and Sampling for Generalized Bergman Spaces on finite Riemann surfaces

Schuster , Alexander; Varolin , Dror
We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerdá, Seip, Wallsten and others, our conditions for interpolation and sampling are as follows: If a certain upper density of the sequence has value less that 1, then the sequence is interpolating, while if a certain lower density has value greater than 1, then the sequence is sampling. Unlike previous works, we introduce a family of densities all of which provide sufficient conditions. Thus we obtain new results even in classical cases, some of which might be useful in industrial applications. The main point...

11. Global infinite energy solutions of the critical semilinear wave equation

Germain , Pierre
We consider the critical semilinear wave equation \begin{equation*} (NLW)_{2^*-1} \;\;\; \left\{ \begin{aligned} \square u + |u|^{2^*-2} u & = 0 \\ u_{|t=0} & = u_0 \\ \partial_t u_{|t=0} & = u_1 \, \,, \end{aligned} \right. \end{equation*} set in $\mathbb{R}^d$, $d \geq 3$, with $2^* = \frac{2d}{d-2} \,\cdotp$ Shatah and Struwe [Shatah, J. and Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics 2. New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, RI, 1998] proved that, for finite energy initial data (ie if $(u_0,u_1) \in \dot{H}^1 \times L^2$), there exists a global solution such that $(u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2)$. Planchon [Planchon, F.: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math....

12. Soluble products of connected subgroups

Gállego , M. Pilar; Hauck , Peter; Pérez-Ramos , M. Dolores
The main result in the paper states the following: For a finite group $G=AB$, which is the product of the soluble subgroups $A$ and $B$, if $\langle a,b \rangle$ is a metanilpotent group for all $a\in A$ and $b\in B$, then the factor groups $\langle a,b \rangle F(G)/F(G)$ are nilpotent, $F(G)$ denoting the Fitting subgroup of $G$. A particular generalization of this result and some consequences are also obtained. For instance, such a group $G$ is proved to be soluble of nilpotent length at most $l+1$, assuming that the factors $A$ and $B$ have nilpotent length at most $l$. Also for any finite soluble group $G$ and $k\geq 1$,...

13. Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds

Desvillettes , Laurent; Fellner , Klemens
In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in $L^1$ to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global $L^{\infty}$ bound via interpolation of a polynomially growing $H^1$ bound with the almost exponential $L^1$ convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.

14. On the number of ovals of a symmetry of a compact Riemann surface

Bujalance , Emilio; Cirre , Francisco Javier; Gamboa , José Manuel; Gromadzki , Grzegorz
Let $X$ be a symmetric compact Riemann surface whose full group of conformal automorphisms is cyclic. We derive a formula for counting the number of ovals of the symmetries of $X$ in terms of few data of the monodromy of the covering $X\rightarrow X/G$, where $G=\mbox{\rm Aut\/}^\pm X$ is the full group of conformal and anticonformal automorphisms of $X$.

15. Erratum: A Parabolic Quasilinear Problem for Linear Growth Functionals (Rev. Mat. Iberoamericana \textbf{18} (2002), no. 1, 135-185)

Andreu , Fuensanta; Caselles , Vicent; Mazón , José Manuel
We give the correct proof of Lemma 3.6 of the paper {\it A Parabolic Quasilinear Problem for Linear Growth Functionals} (Rev. Mat. Iberoamericana {\bf 18} (2002), no. 1, 135-185).

16. Sampling Sets for the Nevanlinna class

Massaneda , Xavier; Thomas , Pascal J.
We propose a definition of sampling set for the Nevanlinna and Smirnov classes in the disk and show its equivalence with the notion of determination set for the same classes. We also show the relationship with determination sets for related classes of functions and deduce a characterization of Smirnov sampling sets. For Nevanlinna sampling we give general conditions (necessary or sufficient), from which we obtain precise geometric descriptions in several regular cases.

17. Bound state solutions for a class of nonlinear Schrödinger equations

Bonheure , Denis; Van Schaftingen , Jean
We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schr�dinger equations of the form  -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $where$V, K$are positive continuous functions and$p > 1$is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential$V$is allowed to vanish at infinity and the competing function$K$does not have to be bounded. In the \emph{semi-classical limit}, i.e. for$\varepsilon\sim 0$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function$\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$, where$\theta=(p+1)/(p-1)-N/2$. A... 18. Comparison of the classical BMO with the BMO spaces associated with operators and applications Deng , Donggao; Duong , Xuan Thinh; Sikora , Adam; Yan , Lixin Let$L$be a generator of a semigroup satisfying the Gaussian upper bounds. A new${\rm BMO}_L$space associated with$L$was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new${\rm BMO}_L$spaces in the theory of singular integration. For example we obtain${\rm BMO}_L$estimates and interpolation results for fractional powers, purely imaginary... 19. Heat kernel transform for nilmanifolds associated to the Heisenberg group Krötz , Bernhard; Thangavelu , Sundaram; Xu , Yuan We study the heat kernel transform on a nilmanifold$M$of the Heisenberg group. We show that the image of$L^2(M)$under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces. 20. Rees algebras on smooth schemes: integral closure and higher differential operator Villamayor U. , Orlando Let$V$be a smooth scheme over a field$k$, and let$\{I_n, n\geq 0\}$be a filtration of sheaves of ideals in$\mathcal{O}_V$, such that$I_0=\mathcal{O}_V$, and$I_s\cdot I_t\subset I_{s+t}$. In such case$\bigoplus I_n$is called a Rees algebra. A Rees algebra is said to be a differential algebra if, for any two integers$N > n$and any differential operator$D$of order$n$,$D(I_N)\subset I_{N-n}\$. Any Rees algebra extends to a smallest differential algebra. There are two extensions of Rees algebras of interest in singularity theory: one defined by taking integral closures, and another by extending the algebra to a differential algebra. We study...

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