A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right‐hand side in Lp (p⩾1)

Accurate modelling of heat transfer in high‐temperature situations requires accounting for the effect of heat radiation. In complex industrial applications involving dissipative heating, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L¹. In this paper, we focus on a stationary heat equation with nonlocal boundary conditions and L^p right‐hand side, with p?1 being arbitrary. Thanks to new coercivity results, we are able to produce energy estimates that involve only the L^p norm of the heat sources and to prove the existence of weak solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Periodic solutions of the Navier–Stokes equations in a perturbed half‐space and an aperture domain

We shall construct a periodic strong solution of the Navier–Stokes equations for some periodic external force in a perturbed half‐space and an aperture domain of the dimension n?3. Our proof is based on L^p–L^q estimates of the Stokes semigroup. We apply L^p–L^q estimates to the integral equation which is transformed from the original equation. As a result, we obtain the existence and uniqueness of periodic strong solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Lp-Lq estimates for a class of pseudo-differential equations and their applications

This paper is concerned with the L ^p -L ^q estimates of the solutions for a class of pseudodifferential equations under some suitable degenerate assumptions. As applications, these estimates can be used to show that a generalized Schr?dinger operator with some integrable potential generates a fractionally integrated group in L ^p (ℝ ⁿ ).     

Some Fourier-type operators for functions on unbounded intervals

In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal systems. We prove necessary and sufficient conditions for the boundedness of these operators in suitable weighted L ^p -spaces, with 1 < p < ∞. Moreover, we give error estimates in weighted L ^p and uniform norms.     

Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

一类带食饵趋向的Beddington-DeAngelis捕食者-食饵扩散模型整体解的有界性

本文研究一类带食饵趋向的Beddington-DeAngelis捕食者-食饵扩散模型,其中食饵趋向性描述的是捕食者对食饵数量变化而产生的一种正向迁移.利用Neumann热半群的L^p-L^q估计和带抛物型方程Moser迭代的L^p估计,获得了该模型经典解的整体有界性.     

Weighted sharp boundedness for multilinear commutators

We obtain sharp estimates for some multilinear commutators related to certain sublinear integral operators. These operators include the Littlewood-Paley operator and Marcinkiewicz operator. As an application, we obtain weighted L ^p (p > 1) inequalities and an L log L-type estimate for multilinear commutators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1419–1431, October, 2007.     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.     

Uniform estimates on multi-linear operators with modulation symmetry

In a previous paper [20] in this series, we gaveL ^p estimates for multi-linear operators given by multipliers which are singular on a non-degenerate subspace of some dimensionk. In this paper, we give uniform estimates when the subspace approaches a degenerate region in the casek = 1, and when all the exponentsp are between 2 and ∞. In particular, we recover the non-endpoint uniform estimates for the bilinear Hubert transform in [12]. Dedicated to Tom Wolff     

An extrapolation theorem for nonlinear approximation and its applications

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint L^p-L^q estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are given to spherical m-widths and m-term approximation of the weighted Besov classes.     

Kernels and best approximations related to the system of ultraspherical polynomials

We study the uniformly bounded orthonormal system of functions

where is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of L^p-norms of the kernels of the system . These estimates enable us to prove Nikol'skiı˘-type inequalities for -polynomials. Next, we prove directly that is a basis in each , where w is an arbitrary A_p-weight function. Finally, we apply these results to get sharp inequalities for the best -approximations in L^q in terms of the best -approximations in . For the trigonometric system such inequalities have been already known.     

One application of Floquet's theory to Lp–Lq estimates for hyperbolic equations with very fast oscillations

In this paper we consider the strictly hyperbolic equation u_tt−λ²(t)b²(t)Δu=0. The coefficient consists of an increasing function λ=λ(t) and a non‐constant periodic function b=b(t). We study the question for the influence of these parts on L_p–L_q decay estimates for the solution of the Cauchy problem. A fairly wide class of equations will be described for which the influence of the oscillating part dominates. This implies, on the one hand, that there exist no L_p–L_q decay estimates and, on the other hand, that the energy estimate from Gronwall's inequality is near to an optimal one. Copyright © 1999 John Wiley & Sons, Ltd.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

Some new estimates for the complex Monge-Ampère equation

In this paper, we consider the complex Monge-Ampère equation posed on a compact K?hler manifold. We show how to get L~p(p ∞) and L∞estimates for the gradient of the solution in terms of the continuity of the right-hand side.     

Estimates of Best Approximation and Fourier Transforms in Integral Metrics

Under some assumptions on a function F and its Fourier transform we prove new estimates of best approximation of F by entire functions of exponential type σ in L_p( ), 1 ≤ p < 2. The proof is based on some inequalities for in L₁( ) which may be treated as generalizations of results of Bausov and Telyakovskii. As an application we obtain exact estimates of best approximation of some infinitely differentiable functions.     

Equazioni quasi ellittiche e spaziL p, θ (ω, δ) (I)

Summary Regularity theorems inL ^{2, θ} (ω, δ) spaces are proved for weak solutions of quasielliptic differential equations. In particular, regularization results are obtained in the class of holder continuous functions (with respect to a suitable metric related to the operator). As a consequence, we obtain results and estimates in L^p andL ^{p, θ} spaces for the solution of the Dirichlet problem.

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Lavoro eseguito nell’ambito del Gruppo di Ricerca n^o 46 del Comitato per la Matematica del C N.R.