The Log-effect for 2 by 2 hyperbolic systems

In the present paper we are interested to extend the Log-effect from wave equations with time-dependent coefficients to 2 by 2 strictly hyperbolic systems t_∂U−A(t)x_∂U=0. Besides the effects of oscillating entries of the matrix A=A(t) and interactions between the entries of A we have to take into consideration the system character itself. We will prove by tools from phase space analysis results about H^∞ well- or ill-posedness. The precise loss of regularity is of interest. Finally, we discuss the cone of dependence property.     

Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

We study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in R×Rd. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator u?A(t)u−ut in suitable Lp spaces.     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.     

Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients

This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach, and the density of the continuous functions in the domain of the Γ-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let ?? be a domain in ? ^N and consider a second order linear partial differential operator A in divergence form on ?? which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup $(e^{-t A_{V}})_{t\geqslant0}$ on L ²(??,dx), whose generator A _V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup $(e^{-t A_{V}})_{t\geqslant0}$ is quasi-L ^p -contractive for 1<p<??. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a^*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a^*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.     

Existence of spectral well-behaved ∗-representations

There are several cases, where an m^∗-seminorm p is defined on a ∗-subalgebra of a given ∗-algebra A. This may lead to the construction of an unbounded ∗-representation of A. Such m^∗-seminorms are called unbounded. Given an unbounded m^∗-seminorm p of a ∗-algebra A, the concept of a p-spectral ∗-representation of A is introduced and studied in connection to well-behaved ∗-representations. More precisely, the existence of (p-) spectral well-behaved ∗-representations is investigated on ∗-algebras and locally convex ∗-algebras in terms of certain properties of Pták function, closely related to hermiticity and C^∗-spectrality of the ∗-subalgebras on which this function is defined. Various examples in diverse classes of locally convex algebras illuminate the elaborated results.     

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

Existence of strong solutions of Pucci extremal equations with superlinear growth in Du

We prove existence of strong solutions of Pucci extremal equations with superlinear growth in Du and unbounded coefficients. We apply this result to establish the weak Harnack inequality for L^p-viscosity supersolutions of fully nonlinear uniformly elliptic PDEs with superlinear growth terms with respect to Du.      

Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II

We construct spike layered solutions for the semilinear elliptic equation −ε²Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K.     

On the stable rank of algebras of operator fields over metric spaces

Let Γ be a finitely generated, torsion-free, two-step nilpotent group. Let C^*(Γ) denote the universal C^*-algebra of Γ. We show that , where for a unital C^*-algebra A, sr(A) is the stable rank of A, and where is the space of one-dimensional representations of Γ. In process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.