Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions

Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditions are considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W1,p regularity for the displacements and Lp regularity for the stresses are proved for some p>2.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Coderivative calculus and metric regularity for constraint and variational systems

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization.     

Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W ^1,p , 1 < p < ∞, regularity. It is proved that such a W ^1,p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.     

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.     

The critical case of clamped thermoelastic systems with interior point control: Optimal interior and boundary regularity results

In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n=dimΩ=2. Indeed, an optimal interior regularity theory was obtained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004] for n=1 and n=3. However, in this reference, an ‘?-loss’ of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces and . The present paper manages to establish that, indeed, one can take ?=0, thus obtaining an optimal interior regularity theory also for the case n=2. The elastic variables have the same interior regularity as in the corresponding elastic problem [R. Triggiani, Regularity with interior point control, Part II: Kirchhoff equations, J. Differential Equations 103 (1993) 394-421] (Kirchhoff). Unlike [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], the present paper establishes the sought-after interior regularity of the thermoelastic problem through a technical analysis based on sharp boundary (trace) regularity theory of Kirchhoff and wave equations. In the process, a new boundary regularity result, not contained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], is obtained for the elastic displacement of the thermoelastic system.     

Regularity criteria for almost every function in Sobolev spaces

In this paper we determine the multifractal nature of almost every function (in the prevalence setting) in a given Sobolev or Besov space according to different regularity exponents. These regularity criteria are based on local Lp regularity or on wavelet coefficients and give a precise information on pointwise behavior.     

2-microlocal analysis of martingales and stochastic integrals

Recently, a new approach in the fine analysis of sample paths of stochastic processes has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of continuous martingales and stochastic integrals. We proved that the almost sure 2-microlocal frontier of a martingale can be obtained through the local regularity of its quadratic variation. It allows to link the Hölder regularity of a stochastic integral to the regularity of the integrand and integrator processes. These results provide a methodology to predict the local regularity of diffusions from the fine analysis of its coefficients. We illustrate our work with examples of martingales with unusual complex regularity behaviour and square of Bessel processes.     

Global properties of invariant measures

We study global regularity properties of invariant measures associated with second order differential operators in R^N. Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.     

Loss of regularity for p-evolution type models

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.     

Discrete time analytic semigroups and the geometry of Banach spaces

We study different notions of discrete maximal regularity for discrete-time abstract Cauchy problems in Banach spaces. First we look at l ₂-discrete maximal regularity and show that Hilbert spaces are the only Banach spaces, among spaces with an unconditional basis, in which the analyticity of the associated discrete-time semigroup is a sufficient condition to obtain this kind of regularity. We then turn to different notions of regularity, in a l ₁ and in a l _∞ sense. We link the existence of particular semigroups such that the associated Cauchy problem has one of these maximal regularities to the geometry of the underlying Banach space (more precisely, to the existence of a complemented subspace isomorphic to c ₀ or l ₁). Finally, we give some elements to compare these regularities.     

Solvability problems of bivariate interpolation I

This paper is devoted to bivariate interpolation. The problem is to find a polynomialP(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matricesE, their regularity, their almost regularity, and finally the regularity of the pairE, Z for a given set of knotsZ. Many concrete examples and applications are possible.     

Regularity ofp-energy minimizing maps andp-superstrongly unstable indices

We derive the first and the second variational formulas forp-energy functional on maps between Riemannian manifolds, obtain a Bochner formula with related estimates and discuss Liouville-type theorems and the regularity ofp-minimizers. In particular, via an extrinsic average variational method,p-superstrongly unstable manifolds and indices are found and their role in the regularity theory is established.     

Characterizations of ɛ-duality gap statements for constrained optimization problems

In this paper we present different regularity conditions that equivalently characterize various ?-duality gap statements (with ? ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ?-subdifferentials. When ? = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.     

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

We prove existence of strongly continuous evolution systems in L² for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data.     

The regularity for nonlinear parabolic systems of p-Laplacian type with critical growth

We study the Hölder regularity of weak solutions to the evolutionary p  -Laplacian system with critical growth on the gradient. We establish a natural criterion for proving that a small solution and its gradient are locally Hölder continuous almost everywhere. Actually our regularity result recovers the classical result in the case p=2$p=2$ [16] and can be applied to study the regularity of the heat flow for m-dimensional H-systems as well as the m-harmonic flow.     

Regularity of Traveling Free Surface Water Waves with Vorticity

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C ^2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C ^2,μ .     

Stability for Parabolic Quasiminimizers

This paper studies parabolic quasiminimizers which are solutions to parabolic variational inequalities. We show that, under a suitable regularity condition on the boundary, parabolic Q-quasiminimizers related to the parabolic p-Laplace equations with given boundary values are stable with respect to parameters Q and p. The argument is based on variational techniques, higher integrability results and regularity estimates in time. This shows that stability does not only hold for parabolic partial differential equations but it also holds for variational inequalities.     

Construction of {\mathcal L}^{{p}}-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$ -strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvent of kernels having the $\mathcal L^p$ -strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.     

Representation of the Singularities of a Function

We show that a functionfhaving a little uniform smoothness can be locally represented at any pointx₀as [equation] wheregis an indefinitely oscillatin function; the value of β of the regularity ofrare related by the 2-microlocal regularity iff.