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In this article we survey recent results on the holomorphic functional calculus for sectorial operators on Banach spaces. Starting from important classes of operators with a bounded H∞-calculus and its essential applications we show how the classical Hilbert space theory of the H∞-calculus can be extended to the Banach space setting. In particular, we discuss characterizations in terms of dilations, interpolation theory and square function estimates. These results lead to perturbation results which in turn allow us to verify the boundedness of the H∞-calculus for new classes of partial differential operators. [ABSTRACT FROM AUTHOR]

We propose a modification of Prohorov's theorem on projective limits of Radon measures which can be directly applied to the construction of Wiener measure on the space of continuous functions, and we give such a construction. [ABSTRACT FROM AUTHOR]

In this short paper, we present a formalism which specifies the notions of consistency and stability of finite element methods for the numerical approximation of nonlinear partial differential equations of elliptic and parabolic type. This formalism can be found in [4], [7], [10], and allows to establish a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods. In concrete cases, the Cléement's interpolation technique [6] is very useful in order to establish local a posteriori error estimates. This paper uses some ideas of [10] and its main goal is to show in a very simple setting, the mathematical arguments which lead to the stability and convergence of Galerkin methods. The bibliography concerning this subject is very large and the references of this paper are no exhaustive character. In order to obtain a large bibliography on the a posteriori error estimates, we report the lecturer to Verfürth's book and its bibliography [12]. [ABSTRACT FROM AUTHOR]

In this paper we investigate the nonlinear Cahn-Hilliard equation with nonconstant temperature and dynamic boundary conditions. We show maximal Lp-regularity for this problem with inhomogeneous boundary data. Furthermore we show global existence and use the Lojasiewicz-Simon inequality to show that each solution converges to a steady state as time tends to infinity, as soon as the potential Φ and the latent heat λ satisfy certain growth conditions. [ABSTRACT FROM AUTHOR]

We study evolution problems of the type esx∂tu+h(∂x)u = f where h is a holomorphic function on a vertical strip around the imaginary axis, and s > 0. If P is a second-order polynomial we give a complete characterization of the spectrum of the parameter-dependent operator λesx + P(∂x) in Lp(ℝ). We show the surprising result that the spectrum is independent of λ whenever arg λ < π. Moreover, we also characterize the spectrum of ∂tesx + P(∂x), and we show that this operator admits a bounded H∞-calculus. Finally, we describe applications to free boundary problems with moving contact lines, and we study the diffusion equation in an angle or a wedge domain with dynamic boundary conditions. Our approach relies on the H∞-calculus for sectorial operators, the concept of R-boundedness, and recent results for the sum of non-commuting operators. [ABSTRACT FROM AUTHOR]

The complex Ginzburg-Landau equation is characterized by complex coefficients with positive real parts in front of both the Laplacian and the nonlinear term. Let k+iβ be the coefficient of the nonlinear term uq−2u with q ≥ 2. If $$ \kappa ^{ - 1} \beta \leqslant 2\sqrt {q - 1} /(q - 2) $$, then the family of solution operators forms a semigroup of (quasi-)contractions on L2(Ω), Ω ⊂ ℝN. Under an additional condition 2 ≤ q ≤ 2 + N/4 the family forms a semigroup of locally Lipschitz operators even if $$ \kappa ^{ - 1} \beta > 2\sqrt {q - 1} /(q - 2) $$ . As a beginning to understand such semigroups the resolvent problem for the equation is developed. [ABSTRACT FROM AUTHOR]

We consider the parabolic evolutionary equation $$ D_t^\beta (u_t - z) + Au = f,{\text{ }}u{\text{(0) = }}y,{\text{ }}u_t (0) = z $$, in continuous interpolation spaces allowing a singularity as t → 0. Here β ∈ (0, 1). We analyze the corresponding trace spaces and show how they depend on β and on the strength of the singularity. The results are applied to the quasilinear equation $$ D_t^\beta (u_t - z) + A(u,u_t )u = f,{\text{ }}u{\text{(0) = }}y,{\text{ }}u_t (0) = z. $$ [ABSTRACT FROM AUTHOR]

We prove a new Fourier multiplier theorem for operator-valued symbols in UMD spaces with property (α) by making simultaneous use of the various good geometric properties of the Banach spaces in question that are available. Our sufficient condition intersects the known Mihlin-Lizorkin and Hörmander type assumptions. [ABSTRACT FROM AUTHOR]

Systems of hierarchically interacting diffusions allow for a rigorous renormalization analysis. By bringing into play the powerful machinery of stochastic analysis, it is possible to obtain a complete classification of the large space-time behavior of these systems into universality classes. The present paper outlines a general renormalization program that is being pursued since ten years and describes four examples where this program has been successfully carried through. The systems under consideration model the evolution of multi-type populations subject to migration and resampling. [ABSTRACT FROM AUTHOR]

Consider the divergence problem with homogeneous Dirichlet data on a Lipschitz domain. Two approaches for its solutions in the scale of Sobolev spaces are presented. The first one is based on Calderón-Zygmund theory, whereas the second one relies on the Stokes equation with inhomogeneous data. [ABSTRACT FROM AUTHOR]

It is shown that the union of a sequence T1, T2. . . of R-bounded sets of operators from X into Y with R-bounds T1, T2. . ., respectively, is R-bounded if X is a Banach space of cotype q, Y a Banach space of type p, and Σk=1/∞ Tk/r < ∞, where r = pq/(q − p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p ≠ q. The power r is sharp. Each Banach space that contains an isomorphic copy of c0 admits operators T1, T2. . . such that ∥Tk∥ = 1/k, k ∈ ℕ, and T1, T2. . . is not R-bounded. Further it is shown that the set of positive linear contractions in an infinite-dimensional Lp is R-bounded only if p = 2. [ABSTRACT FROM AUTHOR]

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on ℝd can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behavior of the corresponding evolution semigroup St can be described in terms of a function (A, B) ↦ d(A; B) ∈ [0, ∞] over pairs of measurable subsets of ℝd. Then $$ \left {\left( {\varphi A,S_t \varphi B} \right)} \right \leqslant e^{ - d(A;B)^2 (4t)^{ - 1} } \varphi A2\varphi B2 $$ for all t > 0 and all ϕA ↦ L2(A), ϕB ∈ L2(B). Moreover StL2(A) ∈ L2(A) for all t > 0 if and only if d(A;Ac) = ∞ where Ac denotes the complement of A. [ABSTRACT FROM AUTHOR]

We consider a second-order elliptic operator k arising from Hamiltonian systems with friction in ℝ2n perturbed by noise. An invariant measure for this operator is μ(dx, dy) = exp(−2H(x, y))dx dy, where H is the Hamiltonian. We study the realization K: H2(ℝ2n, μ) ↦ L2(ℝ2n, μ) of k in L2(ℝ2n, μ), proving that it is m-dissipative and that it generates an analytic semigroup. [ABSTRACT FROM AUTHOR]

We extend to a more general class of diffusion coefficients the results proved in the previous work [6] on uniqueness, ergodicity and strongly mixing property of the invariant measure for some stochastic reaction-diffusion equations, in which the diffusion term is possibly vanishing and the deterministic part is not asymptotically stable. We obtain our results by random time changes and some comparison arguments with Bessel processes. [ABSTRACT FROM AUTHOR]

Several approximation operators followed Philippe Clément's seminal paper in 1975 and are hence known as Clément-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V ⊂ Wk,p(Ω) onto some finite element space Vh ⊂ Wk,p(Ω) and generalize nodal interpolation operators whenever Wk,p(Ω) ⊄ C0(Ω), i.e., when p ≤ n/k for a bounded Lipschitz domain Ω ⊂ ℝn. The original motivation was H2 ⊄ C0(Ω) for higher dimensions n ≥ 4 and hence nodal interpolation is not well defined. Todays main use of the approximation operators is for a reliability proof in a posteriori error control. The survey reports on the class of Clément type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design. [ABSTRACT FROM AUTHOR]

We prove Harnack type inequalities for linear biharmonic equations containing a Kato potential. Various applications to local boundedness, Hölder continuity and universal estimates of solutions for biharmonic equations are presented. [ABSTRACT FROM AUTHOR]

This monograph is devoted to theory and applications of various types of factorizations for matrix and operator functions belonging to different classes. The types of factorizations described in the book appear in several branches of algebra, analysis and applications. Let us mention a few examples. [ABSTRACT FROM AUTHOR]