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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/1804422017