Critical exponents for nonlinear diffusion equations with nonlinear boundary sources

This paper is concerned with the large time behavior of solutions to two types of nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q₀ and the critical Fujita exponent qc for the problems considered, and show that q₀=qc for the multi-dimensional porous medium equation and non-Newtonian filtration equation with nonlinear boundary sources. This is quite different from the known results that q₀<qc for the one-dimensional case.     

Infinite triangular matrices, q-Pascal matrices, and determinantal representations

We use basic properties of infinite lower triangular matrices and the connections of Toeplitz matrices with generating-functions to obtain inversion formulas for several types of q-Pascal matrices, determinantal representations for polynomial sequences, and identities involving the q-Gaussian coefficients. We also obtain a fast inversion algorithm for general infinite lower triangular matrices.     

Complete metric approximation property for q-Araki–Woods algebras

By adapting an ultraproduct technique of Junge and Zeng, we prove that radial completely bounded multipliers on q-Gaussian algebras transfer to q-Araki–Woods algebras. As a consequence, we establish the $w^{?}$-complete metric approximation property for all q-Araki–Woods algebras. We apply the latter result to show that the canonical ultraweakly dense C^?-subalgebras of q-Araki–Woods algebras are always QWEP.     

Displacement convexity of generalized relative entropies

We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the m-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.     

On the inverse problem for three-dimensional potential scattering

We consider a linear perturbation for the wave equation $\text{□u = 0}\text{in}\text{Ω = E}{}^3$ by “repulsive” smooth potentials q(y) that are small at infinity and suitably small (in magnitude). We use a time-dependent approach to prove that the scattering operator S(q) determines uniquenes uniquely the scatterer q (at least in this class). Energy inequalities will play a central role in our discussion.     

On q-Gaussian Integrable Hamiltonian Reductions in Anisentropic Magneto-gasdynamics

Integrable substructure in 2+1-dimensional anisentropic magneto-gasdynamics is investigated via a general elliptic vortex ansatz. The procedure involves introduction of a q-Gaussian density representation. Thermodynamically consistent relations are isolated associated with certain integrable Hamiltonian reductions.     

On some properties of q-Hahn multiple orthogonal polynomials

This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators, as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogonality conditions are considered. An explicit expression of this q-difference equation will be given. Indeed, this q-difference equation relates polynomials with a given degree evaluated at four consecutive non-uniformed distributed points, which makes these polynomials interesting from the point of view of bispectral problems.     

Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).     

Porous media equation and Sobolev inequalities under negative curvature

We study the link between some modified porous media equation and Sobolev inequalities on a Riemannian manifold M whose Ricci curvature tensor is bounded below by a negative constant −ρ. The method used deals with entropy-energy differentiation and follows the way the author got inequalities under nonnegative Ricci curvature assumptions. The key of the proof is the curvature-dimension criterion.     

Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature

We apply the method of [J. Demange, From porous media equation to generalized Sobolev inequalities on a Riemannian manifold, preprint, http://www.lsp.ups-tlse.fr/Fp/Demange/, 2004] and [J. Demange, Porous Media equation and Sobolev inequalities under negative curvature, preprint, http://www.lsp.ups-tlse.fr/Fp/Demange/, 2004], based on the curvature-dimension criterion and the study of Porous Media equation, to the case of a manifold M with strictly positive Ricci curvature. This gives a new way to prove classical Sobolev inequalities on M. Moreover, this enables to improve non-critical Sobolev inequalities as well. As an application, we study the rate of convergence of the solutions of the Porous Media equation to the equilibrium.     

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings

We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form φ=a+q, where a is an additive mapping and q is a quadratic one.     

On cohomology of finitely generated function groups

Let Γ be a non-elementary finitely generated Kleinian group with region of discontinuity Ω. Letq be an integer,q ≥ 2. The group Λ acts on the right on the vector space Π_2q?2 of polynomials of degree less than or equal to 2q ? 2 via Eichler action. We note by A_qq(Ω, Λ) the space of cusp forms for Λ of weight (?2q) and the space of parabolic cohomology classes by PH¹ (Λ, Π2q?2). Bers introduced an anti-linear map $$\beta _q^* :A^q \left( {\Omega ,\Gamma } \right) - - - \to PH^1 \left( {\Gamma ,\Omega _{2q - 2} } \right)$$ . We try to determine the class of Kleinian groups for which the Bers map is surjective. We show that the Bers map is surjective for geometrically finite function groups. We also obtain a characterization of geometrically finite function groups. As an application, we reprove theorems of Maskit on inequalities involving the dimension of the space of cusp forms supported on an invariant component and the dimension of the space of cusp forms supported on the other components for finitely generated function groups. We also show all these inequalities are equalities for geometrically finiteB-groups.     

Inequalities for the gamma and q-gamma functions

We present several sharp inequalities for the classical gamma and q-gamma functions. Some inequalities involve the psi function and its q-analogue. Our results improve, complement, and generalize some known (nonsharp) estimates.     

Two-weight norm inequalities for commutators of potential type integral operators

We give a condition which is sufficient for the two-weight (p,q) inequalities for commutators of potential type integral operators.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

Weak boundedness properties for maximal operators defined on weighted spaces

The purpose of this paper is to obtain characterizations of weak type (1,q) inequalities,q ≥ 1, for maximal operators defined on weighted spaces by means of the corresponding operator acting over Dirac deltas. We present a technical theorem which allows us to obtain characterizations for a pair of weights belonging to the classA ₁ of weights by means of the fractional maximal operator. Analogous results are obtained for the one-sided fractional maximal operator.     

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q′ = Π _p|q p of the modulus q satisfies log q′ ～ log q. Some applications to zero free regions of Dirichlet L-functions and the Pólya and Vinogradov inequalities are indicated.     

Integrability of Green potentials in fractal domains

We proveL ^q -inequalities for the gradient of the Green potential (Gf) in bounded, connected NTA-domains inR ⁿ ,n≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponentq for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.