Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Erik Burman In this note, we prove error estimates in natural norms on the approximation of the boundary data in the elliptic Cauchy problem, for the finite element method first analysed in E. Burman, Error estimates for stabilised finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I 352 (7–8) (2014) 655–659.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Luca Baracco, Tran Vu Khanh, Stefano Pinton We show here a “weak” Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge–Ampère equation with data in the L p space and Ω satisfying an f-property. The f-property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Angelo R.F. de Holanda In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in R N . We explore fixed point arguments applied to suitable integral equations to get solutions.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Duván Cardona, Michael Ruzhansky In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Vincent Cossart, Olivier Piltant, Bernd Schober Let X be an excellent scheme; we denote by H X the modified Hilbert–Samuel function. This function is upper semi-continuous along X and does not increase for the lexicographical ordering after permissible blowing ups. When X is embedded in a regular ambient scheme, for all x ∈ X , the “stable τ at x” (“τ stable de x”), denoted by τ st ( x ) , is the codimension of the ridge of the tangent cone of X at x in the tangent cone of W at x. It is well known that the function ι : X → N N × − N , x ↦ ( H X ( x ) , − τ st ( x ) ) , does not increase for the lexicographical ordering after permissible blowing ups. In this note, we show that ι is upper semi-continuous along X . This result is generalized to the non-embedded case.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Indranil Biswas, Subramaniam Senthamarai Kannan, Donihakalu Shankar Nagaraj Let G ‾ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over C . Let T ‾ ⊂ G ‾ be the closure of a maximal torus T ⊂ G . We prove that the group of all automorphisms of the variety T ‾ is the semi-direct product N G ( T ) ⋊ D , where N G ( T ) is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if G ≠ PSL ( 2 , C ) . Note that if G = PSL ( 2 , C ) , then T ‾ = C P 1 and so in this case Aut ( T ‾ ) = PSL ( 2 , C ) .

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Yuri Berest, Alimjon Eshmatov, Wai-Kit Yeung In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in R 3 , we define a differential graded (DG) k-category A ˜ L with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of a distinguished object of A ˜ L coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng, and Sullivan in [13]. The input of our construction is a natural action of the braid group B n on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson, and Vilonen in [19]. As an application, we show that the category of finite-dimensional representations of the link k-category A ˜ L = H 0 ( A ˜ L ) defined as the 0-th homology of A ˜ L is equivalent to the category of perverse sheaves on R 3 that are singular along the link L. We also obtain several generalizations of the category A ˜ L by extending the Gelfand–MacPherson–Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4].

Abstract: Publication date: Available online 6 March 2017 Source:Comptes Rendus Mathematique Author(s): Nam Q. Le We show that for an L 2 drift b in two dimensions, if the Hardy norm of div b is small, then the weak solutions to Δ u + b ⋅ ∇ u = 0 have the same optimal Hölder regularity as in the case of divergence-free drift, that is, u ∈ C loc α for all α ∈ ( 0 , 1 ) .

Abstract: Publication date: Available online 1 March 2017 Source:Comptes Rendus Mathematique Author(s): John Bergdall, Robert Pollack We give a sufficient condition, namely “Buzzard irregularity”, for there to exist a cuspidal eigenform which does not have integral p-adic slope.

Abstract: Publication date: Available online 24 February 2017 Source:Comptes Rendus Mathematique Author(s): Oleg Makarenkov We offer a simple proof of the Lyapunov finite-time stability theorem for Filippov systems which does not use any generalized derivatives to differentiate the composition of the Lyapunov function with absolutely continuous solutions.

Abstract: Publication date: Available online 20 February 2017 Source:Comptes Rendus Mathematique Author(s): Guillaume Legendre, Gabriel Turinici The time discretization of gradient flows in metric spaces uses variants of the celebrated implicit Euler-type scheme of Jordan, Kinderlehrer, and Otto [9]. We propose in this Note a different approach, which allows us to construct two second-order in time numerical schemes. In a metric space framework, we show that the schemes are well defined and prove the convergence for one of them under some regularity assumptions. For the particular case of a Fokker–Planck gradient flow in the Wasserstein space, we obtain (theoretically and numerically) the second-order convergence.

Abstract: Publication date: Available online 20 February 2017 Source:Comptes Rendus Mathematique Author(s): Salah El Ouadih, Radouan Daher Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space L 2 ( α , β ) .

Abstract: Publication date: Available online 20 February 2017 Source:Comptes Rendus Mathematique Author(s): Pu Zhang Let M be the Hardy–Littlewood maximal function and b be a locally integrable function. Denote by M b and [ b , M ] the maximal commutator and the (nonlinear) commutator of M with b. In this paper, the author considers the boundedness of M b and [ b , M ] on Lebesgue spaces and Morrey spaces when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

Abstract: Publication date: February 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 2 Author(s): Sylvain Gandon, Sepideh Mirrahimi In this note, we characterize the solution to a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems arisen from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows us to go further than the Gaussian approximation commonly used by biologists, and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.

Abstract: Publication date: February 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 2 Author(s): Leila Azem, Olivier Pantz We consider a brittle elastic solid (prone to develop fractures) as the limit of a damage model and propose a numerical method to determine its quasi-static evolution in the spirit of Francfort and Marigo [3] and Allaire et al. [1,2].

Abstract: Publication date: February 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 2 Author(s): Annelies Jaspers Inspired by the motivic monodromy conjecture, Halle and Nicaise defined the global monodromy property for Calabi–Yau varieties over a discretely valued field. In this note, we discuss this property for K3 surfaces allowing a strict normal crossings model where no three components in the special fiber have a common intersection. The main result is that the global monodromy property holds for a K3 surface with a so-called flowerpot degeneration. It also holds for K3 surfaces with a chain degeneration under certain conditions.

Abstract: Publication date: February 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 2 Author(s): Bastien Rossetti We give conditions under which the intersection between two attracting immediate basins boundaries of a rational map contains at least one periodic point.

Abstract: Publication date: Available online 16 February 2017 Source:Comptes Rendus Mathematique Author(s): Sylvie Corteel, Jang Soo Kim, Karola Mészáros The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector ( 1 , 0 , … , 0 , − 1 ) . The normalized volume of the Chan–Robbins–Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector ( 1 , 1 , 0 , … , 0 , − 2 ) . We show that the volume of this polytope is a certain power of 2 times the product of consecutive Catalan numbers. Our proof uses constant-term identities and further deepens the combinatorial mystery of why these numbers appear. In addition, we introduce two more families of flow polytopes whose volumes are given by product formulas.

Abstract: Publication date: Available online 13 February 2017 Source:Comptes Rendus Mathematique Author(s): Zoubida Mghazli, Ilyas Naji We present in this note a posteriori error estimates based on the postprocessing technique for the reduced model of flow in fractured porous media, introduced and analysed by V. Martin, J. Jaffré, and J. Roberts. This model is approximated by the Raviart–Thomas finite elements of lowest order. In this type of approximation, the velocity is well approximated. A postprocessing of the pressure appears to be necessary since it does not belong to H 0 1 ( Ω ) . We give an upper bound for the error in the energy norm, with some indicators that are expressed in terms of the reconstruction of the pressure. Numerical results show that all indicators converge to zero when the mesh size goes to zero, with the same speed as the error. One of these indicators can be interpreted as both a discretization indicator and an indicator of the reduced model validity.

Abstract: Publication date: Available online 11 February 2017 Source:Comptes Rendus Mathematique Author(s): Xiaopeng Zhao This paper discusses the large-time behavior of solutions for a new Hall–MHD system in R 3 . Using the Fourier splitting method, we establish the upper bound of the time-decay rate in L 2 ( R 3 ) for weak solutions.

Abstract: Publication date: Available online 10 February 2017 Source:Comptes Rendus Mathematique Author(s): Ali Chouria, Jean-Gabriel Luque We introduce partial r-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.

Abstract: Publication date: Available online 10 February 2017 Source:Comptes Rendus Mathematique Author(s): Farman Mamedov, Sayali Mammadli In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator H υ , ω f ( x ) = υ ( x ) ∫ 0 x f ( t ) ω ( t ) d t to be compactly acting from L p ( ⋅ ) ( 0 , ∞ ) to L q ( ⋅ ) ( 0 , ∞ ) .

Abstract: Publication date: Available online 9 February 2017 Source:Comptes Rendus Mathematique Author(s): François Genoud In this note, we discuss the global dynamics of an integrable nonlocal NLS on R , which has been the object of a recent investigation by integrable systems methods. We prove two results that are in striking contrast with the case of the local cubic focusing NLS. First, finite-time blow-up solutions exist with arbitrarily small initial data in H s ( R ) , for any s ⩾ 0 . On the other hand, the solitons of the local NLS, which are also solutions to the nonlocal equation, are unstable by blow-up for the latter.

Abstract: Publication date: Available online 9 February 2017 Source:Comptes Rendus Mathematique Author(s): Paul Gassiat We give an example of a stochastic Hamilton–Jacobi equation d u = H ( D u ) d ξ which has an infinite speed of propagation as soon as the driving signal ξ is not of bounded variation.

Abstract: Publication date: Available online 9 February 2017 Source:Comptes Rendus Mathematique Author(s): Richard Gratwick, Aidys Sedipkov, Mikhail Sychev, Aris Tersenov In this paper, we prove that if L ( x , u , v ) ∈ C 3 ( R 3 → R ) , L v v > 0 and L ≥ α v + β , α > 0 , then all problems (1), (2) admit solutions in the class W 1 , 1 [ a , b ] , which are in fact C 3 -regular provided there are no pathological solutions to the Euler equation (5). Here u ∈ C 3 [ c , d [ is called a pathological solution to equation (5) if the equation holds in [ c , d [ , u ˙ ( x ) → ∞ as x → d , and ‖ u ‖ C [ c , d ] < ∞ . We also prove that the lack of pathological solutions to the Euler equation results in the lack of the Lavrentiev phenomenon, see Theorem 9; no growth assumptions from below are required in this result.

Abstract: Publication date: Available online 8 February 2017 Source:Comptes Rendus Mathematique Author(s): Christophe Bovet, Pierre Gosselet, Nicole Spillane Preconditioned Krylov subspace methods [7] are powerful tools for solving linear systems but sometimes they converge very slowly, and often after a long stagnation. A natural way to fix this is by enlarging the space in which the solution is computed at each iteration. Following this idea, we propose in this note two multipreconditioned algorithms: multipreconditioned orthomin and multipreconditioned biCG, which aim at solving general nonsingular linear systems in a small number of iterations. After describing the algorithms, we illustrate their behaviour on systems arising from the FETI domain decomposition method, where in order to enlarge the search space, each local component in the usual preconditioner is kept as a separate preconditioner.

Abstract: Publication date: Available online 7 February 2017 Source:Comptes Rendus Mathematique Author(s): Gleb Nenashev In this note we study ideals generated by generic forms in polynomial rings over any algebraicly closed field of characteristic zero. We prove for many cases that the ( d + k ) -th graded component of an ideal generated by generic forms of degree d has the expected dimension (given by dimension count). And as a consequence of our result, we obtain that ideals generated by several generic forms of degrees d usually have the expected Hilbert series. The precise form of this expected Hilbert series, in general, is known as Fröberg's conjecture.

Abstract: Publication date: Available online 6 February 2017 Source:Comptes Rendus Mathematique Author(s): Chandrasheel Bhagwat, A. Raghuram This is an announcement of certain rationality results for the critical values of the degree-2n L-functions attached to GL 1 × SO ( n , n ) over Q for an even positive integer n. The proof follows from studying the rank-one Eisenstein cohomology for SO ( n + 1 , n + 1 ) .

Abstract: Publication date: Available online 6 February 2017 Source:Comptes Rendus Mathematique Author(s): Miguel Escobedo An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in [5] about the existence of global solutions in the so-called self-similar case is not satisfied. The solution is local and blows up in finite time.

Abstract: Publication date: Available online 6 February 2017 Source:Comptes Rendus Mathematique Author(s): Jean Dolbeault, Maria J. Esteban, Michael Loss, Matteo Muratori We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.

Abstract: Publication date: Available online 4 February 2017 Source:Comptes Rendus Mathematique Author(s): Nils Caillerie We study a random process on R n moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale ( t , x , v ) → ( t ε , x ε , v ) , with ε > 0 , before proceeding to a Hopf–Cole transform, which gives a kinetic equation on a potential. We show convergence as ε → 0 of the potential towards the viscosity solution to a Hamilton–Jacobi equation ∂ t φ + H ( ∇ x φ ) = 0 where the Hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies.

Abstract: Publication date: Available online 4 February 2017 Source:Comptes Rendus Mathematique Author(s): Diego Izquierdo Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate–Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate–Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K).

Abstract: Publication date: Available online 1 February 2017 Source:Comptes Rendus Mathematique Author(s): Juliette Chabassier, Sébastien Imperiale This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.

Abstract: Publication date: Available online 1 February 2017 Source:Comptes Rendus Mathematique Author(s): Victor G. Kac, Minoru Wakimoto We point out that it is immediate by our character formula that in the case of a boundary level the characters of admissible representations of affine Kac–Moody algebras and the corresponding W-algebras decompose in products in terms of the Jacobi form ϑ 11 ( τ , z ) .

Abstract: Publication date: Available online 18 January 2017 Source:Comptes Rendus Mathematique Author(s): Len Meas In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain Ω ⊂ R 3 with a smooth boundary ∂ Ω ≠ ∅ . Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [1,2]. Better estimates in strictly convex domains have been obtained in [4]. Our case of cylindrical domains is an extension of the result of [4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.

Abstract: Publication date: Available online 18 January 2017 Source:Comptes Rendus Mathematique Author(s): Jean Baptiste Patenou In this paper, one considers a Cauchy problem with data on a characteristic cone for the Einstein–Vlasov system in temporal gauge. One highlights gauge-dependent constraints that, supplemented by the standard constraints i.e. the Hamiltonian and the momentum constraints, define the full set of constraints for the considered setting. One studies their global resolution from a suitable choice of some free data, the behavior of the deduced initial data at the vertex of the cone, and the preservation of the gauge.

Abstract: Publication date: Available online 17 January 2017 Source:Comptes Rendus Mathematique Author(s): Xiao-Lei Liu, Sheng-Li Tan We obtain a uniform bound for the effective Bogomolov conjecture, which depends only on the genus g of the curve. The bound grows as O ( g − 3 ) as g tends to infinity.

Abstract: Publication date: Available online 16 January 2017 Source:Comptes Rendus Mathematique Author(s): Sen Yang By using the infinitesimal methods due to Bloch, Green, and Griffiths in [1,4], we construct an infinitesimal form of the regulator map and verify that its kernel is Ω C / Q 1 , which suggests that Question 1.1 seems reasonable at the infinitesimal level.

Abstract: Publication date: Available online 28 December 2016 Source:Comptes Rendus Mathematique Author(s): Anton Alekseev, Nariya Kawazumi, Yusuke Kuno, Florian Naef We define a family KV ( g , n + 1 ) of Kashiwara–Vergne problems associated with compact connected oriented 2-manifolds of genus g with n + 1 boundary components. The problem KV ( 0 , 3 ) is the classical Kashiwara–Vergne problem from Lie theory. We show the existence of solutions to KV ( g , n + 1 ) for arbitrary g and n. The key point is the solution to KV ( 1 , 1 ) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman–Turaev Lie bialgebra g ( g , n + 1 ) . In more detail, we show that every solution to KV ( g , n + 1 ) induces a Lie bialgebra isomorphism between g ( g , n + 1 ) and its associated graded gr g ( g , n + 1 ) . For g = 0 , a similar result was obtained by G. Massuyeau using the Kontsevich integral. For g ≥ 1 , n = 0 , our results imply that the obstruction to surjectivity of the Johnson homomorphism provided by the Turaev cobracket is equivalent to the Enomoto–Satoh obstruction.

Abstract: Publication date: Available online 27 December 2016 Source:Comptes Rendus Mathematique Author(s): Piermarco Cannarsa, Wei Cheng, Albert Fathi We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.

Abstract: Publication date: Available online 27 December 2016 Source:Comptes Rendus Mathematique Author(s): Xuegang Hu, Liangchen Wang, Chunlai Mu, Ling Li The quasilinear chemotaxis–haptotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) u t = + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R 3 . Here χ > 0 , ξ > 0 and μ > 0 , D ( u ) ≥ c D u m − 1 for all u > 0 with some c D > 0 and D ( u ) > 0 for all u ≥ 0 . It is shown that if the ratio χ μ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.

Abstract: Publication date: Available online 27 December 2016 Source:Comptes Rendus Mathematique Author(s): Byeong Moon Kim, Poo-Sung Park We give a lower bound for class numbers of unimodular ternary Hermitian lattices over imaginary quadratic fields. This shows that class numbers of unimodular Hermitian lattices grow infinitely as the field discriminants grow.

Abstract: Publication date: Available online 27 December 2016 Source:Comptes Rendus Mathematique Author(s): Mayukh Mukherjee We use the real analyticity of the Ricci flow with respect to time proved by B. Kotschwar to extend a result of P. Buser, namely, we prove that the Laplace spectra of negatively curved compact orientable surfaces having the same genus γ ≥ 2 , the same area and the same curvature bounds vary in a “controlled way”, of which we give a quantitative estimate in our main theorem. The basic technical tool is a variational formula that provides the derivative of an eigenvalue branch under the normalized Ricci flow. In a related manner, we also observe how the above-mentioned real analyticity result can lead to unexpected conclusions concerning the spectral properties of generic metrics on a compact surface of genus γ ≥ 2 .

Abstract: Publication date: Available online 21 December 2016 Source:Comptes Rendus Mathematique Author(s): Davide Barbieri, Eugenio Hernández, Azita Mayeli Given a lattice Λ in a locally compact Abelian group G and a measurable subset Ω with finite and positive measure, then the set of characters associated with the dual lattice form a frame for L 2 ( Ω ) if and only if the distinct translates by Λ of Ω have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.