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On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations
https://ddd.uab.cat/record/180449
For scalar fully nonlinear partial differential equations F(x, D2u(x)) = 0 with x ∈ Ω b RN , we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem, where F(x, ·) need not be monotone on all of S(N), the space of symmetric N ×N matrices. We treat admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [20]and extend the program initiated by Harvey and Lawson [11] in the homogeneous case when F does not depend on x. In particular, for the set valued map Θ defining the elliptic branch by way of the differential inclusion D2u(x) ∈ ∂Θ(x), we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron's method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by Θ. Structural conditions on F are then derived which ensure the existence of an elliptic map Θ with the needed regularity. Concrete applications are given in which standard structuralconditions on F may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge–Ampère equations and equations prescribing eigenvalues of the Hessian. Cirant, MarcoMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804492017A note on covers of fibred hyperbolic manifolds
https://ddd.uab.cat/record/180448
For each surface S of genus g > 2 we construct pairs of conjugate pseudo-Anosov maps, ϕ1 and ϕ2, and two non-equivalent covers pi : S˜ −→ S, i = 1, 2, so that the lift of ϕ1 to S˜ with respect to p1 coincides with one of ϕ2 with respect to p2. The mapping tori of the ϕi and their lift provide examples of pairs of hyperbolic 3-manifolds so that the first is covered by the second in two different ways. Los, JérômeMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804482017Sur la séparation des caractères par les Frobenius
https://ddd.uab.cat/record/180447
In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK. We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of factorization in Fp[X], where P ∈ Z[X] is a monic, irreducible polynomial of squarefree discriminant; (ii) on the estimation of the maximal number of tamely ramified extensions of Galois group An over a fixed number field K. To finish, we discuss some statistics in the quadratic number fields case (real and imaginary) concerning the separation of two irreducible unramified characters of the alterning group An,for n = 5, 7, 13. Euvrard, CharlotteMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804472017Examples of finite-dimensional hopf algebras with the dual Chevalley property
https://ddd.uab.cat/record/180446
We present new Hopf algebras with the dual Chevalley property by determining all semisimple Hopf algebras Morita-equivalent to a group algebra over a finite group, for a list of groups supporting a non-trivial finite-dimensional Nichols algebra. Andruskiewitsch, NicolasMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804462017Weighted Lp estimates of Kato square roots associated to degenerate elliptic operators
https://ddd.uab.cat/record/180445
Let w be a Muckenhoupt A2(Rn) weight and Lw := −w−1 div(A∇) the degenerate elliptic operator on the Euclidean space Rn, n ≥ 2. In this article, the authors establish some weighted Lp estimates of Kato square roots associated to the degenerate elliptic operators Lw. More precisely, the authors prove that, for w ∈ Ap(Rn), p ∈ (2n n+1 , 2] and any f ∈ C∞c (Rn), kL 1/2 w (f)kLp(w,Rn) ∼ k∇fkLp(w,Rn), where C∞c (Rn) denotes the set of all infinitely differential functions with compact supports and the implicit equivalent positive constants are independent of f. Yang, DachunMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804452017Irreducible representations of knot groups into SL(n, C)
https://ddd.uab.cat/record/180444
The aim of this article is to study the existence of certain reducible, metabelian representations of knot groups into SL(n, C) which generalize the representations studied previously by G. Burde and G. de Rham. Under specific hypotheses we prove the existence of irreducible deformations of such representations of knot groups into SL(n, C). Ben Abdelghani, LeilaMon, 04 Sep 2017 05:59:49 GMThttps://ddd.uab.cat/record/1804442017Separation in the BNSR-invariants of the pure braid groups
https://ddd.uab.cat/record/180443
We inspect the BNSR-invariants Σm(Pn) of the pure braid groups Pn, using Morse theory. The BNS-invariants Σ1 (Pn) were previously computed by Koban, McCammond, and Meier. We prove that for any 3≤m≤n, the inclusion Σm−2 (Pn) ⊆ Σm−3 (Pn) is proper, but Σ∞(Pn) = Σn−2 (Pn). We write down explicit character classes in each relevant Σm−3 (Pn)\Σm−2 (Pn). In particular we get examples of normalsubgroups N ≤ Pn with Pn/N ∼= Z such that N is of type Fm−3 but not Fm−2, for all 3 ≤ m ≤ n. Zaremsky, Matthew C. B.Mon, 04 Sep 2017 05:59:48 GMThttps://ddd.uab.cat/record/1804432017Sobolev regularity of the Beurling transform on planar domains
https://ddd.uab.cat/record/180442
Consider a Lipschitz domain Ω and the Beurling transform of its characteristic function BχΩ(z) = −p. v. 1 πz2 ∗ χΩ(z). It is shown that if the outward unit normal vector N of the boundary of the domain is in the trace space of Wn,p(Ω) (i. e. , the Besov space Bn−1/p p,p (∂Ω)) then BχΩ ∈ Wn,p(Ω). Moreover, when p > 2 the boundedness of the Beurling transform on Wn,p(Ω) follows. This fact has farreaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation. Prats Soler, MartíMon, 04 Sep 2017 05:59:48 GMThttps://ddd.uab.cat/record/1804422017Right Engel elements of stability groups of general series in vector spaces
https://ddd.uab.cat/record/168353
Let V be an arbitrary vector space over some division ring D, L a general series of subspaces of V covering all of V \ {0} and S the full stability subgroup of L in GL(V). We prove that always the set of bounded right Engel elements of S is equal to the w-th term of the upper central series of S and that the set of right Engel elements of S is frequently equal to the hypercentre of S. Wehrfritz, B. A. F.Mon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683532017Localization genus
https://ddd.uab.cat/record/168352
Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We dene in fact the notion of localization genus for any homotopical localization functor in the sense of Bouseld and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization. Møller, Jesper M.Mon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683522017Continuity of solutions to space-varying pointwise linear elliptic equations
https://ddd.uab.cat/record/168351
We consider pointwise linear elliptic equations of the form Lα uα = ŋα on a smooth compact manifold where the operators Lα are in divergence form with real, bounded, measurable coefficients that vary in the space variableα. We establish L2-continuity of the solutions at α whenever the coefficients of Lα are L∞ -continuous at α and the initial datum is L2 -continuous at α. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics gt that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on ʍ with a C1 heat kernel on a “non-singular" nonempty open subset Ɲ, we show that α à gt (α) is continuous whenever α € Ɲ. Bandara, LashiMon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683512017Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian
https://ddd.uab.cat/record/168350
Della Pietra, FrancescoMon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683502017Equigeneric and equisingular families of curves on surfaces
https://ddd.uab.cat/record/168349
We investigate the following question: let C be an integral curve contained in a smooth complex algebraic surface X; is it possible to deform C in X into a nodal curve while preserving its geometric genus? We armatively answer it in most cases when X is a Del Pezzo or Hirzebruch surface (this is due to Arbarello and Cornalba, Zariski, and Harris), and in some cases when X is a K3 surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative. Dedieu, ThomasMon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683492017An example pertaining to the failure of the Besicovitch-Federer structure theorem in Hilbert space
https://ddd.uab.cat/record/168348
We give an example, in the innite dimensional separable Hilbert space, of a purely unrectiable Borel set with nite nonzero one dimensional Hausdor measure, whose projection is nonnegligible in a set of directions which is not Aronszajn null. De Pauw, ThierryMon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683482017Low energy canonical immersions into hyperbolic manifolds and standard spheres
https://ddd.uab.cat/record/168347
We consider critical points of the global L2-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of the first of the functionals mentioned. del Rio, HebertoMon, 19 Dec 2016 05:29:45 GMThttps://ddd.uab.cat/record/1683472017A formula for the Dubrovnik polynomial of rational knots
https://ddd.uab.cat/record/168346
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system. Caprau, CarmenMon, 19 Dec 2016 05:29:44 GMThttps://ddd.uab.cat/record/1683462017A Marcinkiewicz integral type characterization of the Sobolev space
https://ddd.uab.cat/record/168345
In this paper we present a new characterization of the Sobolev space W1,p , 1 < p < ∞ which is a higher dimensional version of a result of Waterman [32]. We also provide a new and simplified proof of a recent result of Alabern, Mateu, and Verdera [2]. Finally, we generalize the results to the case of weighted Sobolev spaces with respect to a Muckenhoupt weight. Hajlasz, PiotrMon, 19 Dec 2016 05:29:44 GMThttps://ddd.uab.cat/record/1683452017Automorphisms groups of simplicial complexes of infinite type surfaces
https://ddd.uab.cat/record/168344
Let S be an orientable surface of innite genus with a nite numberof boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S), and the Schmutz graph G(S) of S. When all topological ends of S carry genus, we show that all elements in the automorphismgroups Aut(C(S)), Aut(N(S)), and Aut(G(S)) are geometric, i. e. these groups are naturally isomorphic to the extended mapping class group MCG(S) of the innite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S)). Hernández Hernández, JesúsMon, 19 Dec 2016 05:29:44 GMThttps://ddd.uab.cat/record/1683442017Bilinear weighted Hardy inequality for nonincreasing functions
https://ddd.uab.cat/record/168343
Krepela, MartinMon, 19 Dec 2016 05:29:44 GMThttps://ddd.uab.cat/record/1683432017Common zeros preserving maps on vector-valued function spaces and Banach modules
https://ddd.uab.cat/record/160583
Let X, Y be Hausdorff topological spaces, and let E and F be Hausdorff topological vector spaces. For certain subspaces A (X,E) and A(Y, F) of C(X,E) and C(Y, F) respectively (including the spaces of Lipschitz functions), we characterize surjections S, T : A (X;E) → A(Y, F), not assumed to be linear, which jointly preserve common zeros in the sense that Z (f – f’) ∩ Z (f – f’) ∩ Z (g – g’) ≠ 0 if and only if Z (Sf – Sf’) ∩ Z (Tg – Tg´) ≠ 0 for all f, f’, g, g’ ∈ A (X, E). Here Z (·)denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective linear maps which jointly preserve common zeros in module case. Hosseini, MalihehThu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605832016Convergence of functions of self-adjoint operators and applications
https://ddd.uab.cat/record/160582
The main result (roughly) is that if Hi converges weakly to H and if also f (Hi) converges weakly to f(H), for a single strictly convex continuous function f, then (Hi) must converge strongly to H. One application is that if f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then M must be invariant for H. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9. 4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If h and f(h) are both quasimultipliers, then h must be a multiplier. Also (still roughly stated), if h and f(h) are both in pAsap, for a closed projection p, then h must be strongly q-continuous on p. Brown, Lawrence G.Thu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605822016Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings
https://ddd.uab.cat/record/160581
We consider a family of dissipative active scalar equations outside the L2-space. This was introduced in [7] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time-decay of solutions, with-out smallness assumptions, for initial data belonging to the critical Lebesgue space Ln/2y-β (ℝn ) which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators. Ferreira, Lucas C. F.Thu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605812016Röver's simple group is of type F ∞
https://ddd.uab.cat/record/160580
We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimplicial complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity. Belk, JamesThu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605802016Integral restriction for bilinear operators
https://ddd.uab.cat/record/160579
We consider the integral domain restriction operator TΩ for certain bilinear operator T. We obtain that if (s, p1, p2) satisfies 1 p1+ 1 p2 ≥ 2 min{1,s} and kTkLp1 ×Lp2→Ls < ∞, then kTΩkLp1 ×Lp2→Ls < ∞. For some special domain Ω, this property holds for triplets (s, p1, p2) satisfying 1 p1 + 1 p2 > 1 min{1,s}. Zhao, WeirenThu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605792016The Kato Square Root Problem follows from an extrapolation property of the Laplacian
https://ddd.uab.cat/record/160578
On a domain Ω ⊆ _ Rd we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain H1/0 (Ω) ⊆ V ⊆ H1 (Ω). Under very mild assumptions on Ω and V we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains. Egert, MoritzThu, 14 Jul 2016 06:36:06 GMThttps://ddd.uab.cat/record/1605782016