Holomorphic Semigroup Research Articles

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

Asymptotic Behavior of Solutions of a Complete Second-Order Integro-Differential Equation

In this paper, we study a complete second-order integro-differential operator equation in a Hilbert space. The difference-type kernel of an integral perturbation is a holomorphic semigroup bordered by unbounded operators. The asymptotic behavior of solutions of this equation is studied. Asymptotic formulas for solutions are proved in the case when the right-hand side is close to an almost periodic function. The obtained formulas are applied to the study of the problem of forced longitudinal vibrations of a viscoelastic rod with Kelvin-Voigt friction.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Holomorphic semigroups of finite shift in the unit disc

We give three necessary and sufficient conditions so that a parabolic holomorphic semigroup (ϕt) in the unit disc is of finite shift. One is in terms of the asymptotic behavior of speeds of convergence, the second one is related to the hyperbolic metric of its Koenigs domain Ω and the latter one deals with Euclidean properties of Ω.

Comparision of Classical Holomorphic Dynamics and Holomorphic Semigroup Dynamics-I

In this paper, we compare existing results of the classical holomorphic dynamics with some of the existing results of holomorphic semigroup dynamics. We also prove some results of holomorphic semigroup dynamics, and we see whether there is a connection or contrast with classical one. Also, we see how far the results generalize, and what new phenomena appear.

A Factorization of a Quadratic Pencils of Accretive Operators and Applications

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. We establish a criterion in order that the linear factors, into which the pencil splits, generates a holomorphic semi-group of contraction operators. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second-order differential equation in the non-homogeneous case. An illustrative example is also given.

Perturbation by weakly continuous forms and semigroups on Hardy space

In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).

On sectoriality of degenerate elliptic operators

AbstractLet $c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all $k,l \in \{1, \ldots , d\};$ and $\Omega \subset \mathbb{R}^{d}$ be open with uniformly $C^{2}$ boundary. We consider the divergence form operator $A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in $L_p(\Omega )$ when the coefficient matrix satisfies $(C(x) \xi , \xi ) \in \Sigma _\theta$ for all $x \in \Omega$ and $\xi \in \mathbb{C}^{d}$, where $\Sigma _\theta$ be the sector with vertex 0 and semi-angle $\theta$ in the complex plane. We show that a sectorial estimate holds for $A_p$ for all $p$ in a suitable range. We then apply these estimates to prove that the closure of $-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.

A Comparison of Classical Holomorphic Dynamics and Holomorphic Semigroup Dynamics-II

In this paper, we prove that the escaping set of a transcendental semi group is S-forward invari-ant. We also prove that if a holomorphic semi group is a belian, then the Fatou, Julia, and escaping sets are S-completely invariant. We also investigate certain cases and conditions that the holomorphic semi group dynamics exhibits the similar dynamical behavior just like a classical holomorphic dynamics. Frequently, we also examine certain amount of connections and contrasts between classical holomorphic dynamics and holomorphic semi group dynamics.

Ill-posed problems associated with sectorial operators in Banach space

Ill-posed problems of the form u′(t)=Au+h, 0<t<T, u(0)=φ are studied in Banach space under two related scenarios, both of which consider a well-posed logarithmic approximation. In the first scenario, −A generates a bounded holomorphic semigroup of angle θ=π2, and so A is sectorial of angle 0. In the second, where θ may be relaxed to any value in (0,π2], we impose an additional assumption that e−A is sectorial. Regularization results for the original ill-posed problem are then established in the linear case where h=h(t), and subsequently extended to nonlinear problems where h is replaced by h(t,u(t)). We compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations, both linear and nonlinear in Lp(Rn), 1<p<∞.

Fatou, Julia, and escaping sets of conjugate holomorphic semigroups

We define commutator of a holomorphic semigroup, and on the basis of this concept, we define conjugate semigroups of a holomorphic semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given holomorphic semigroup is nearly abelian. We also prove that image of each of Fatou, Julia, and escaping sets of a holomorphic semigroup under commutator (affine complex conjugating map) is equal respectively, to the Fatou, Julia, and escaping sets of the conjugate semigroup. Finally, we prove that every element of a nearly abelian holomorphic semigroup S can be written as the composition of an element from the set generated by the set of commutators !(S) and the composition of the certain powers of its generators..

On p-elliptic divergence form operators and holomorphic semigroups

Second order divergence form operators are studied on an open set with various boundary conditions. It is shown that the p-ellipticity condition of Carbonaro-Dragicevic and Dindos-Pipher implies extrapolation to a holomorphic semigroup on Lebesgue spaces in a p-dependent range of exponents that extends the maximal range for general strictly elliptic coefficients. Results have immediate consequences for the harmonic analysis of such operators, including Hoo-calculi and Riesz transforms.

Asymptotic behavior of orbits of holomorphic semigroups

Let (ϕt) be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let Ω be the starlike at infinity domain image of the Koenigs function of (ϕt). In this paper we characterize the type of convergence of the orbits of (ϕt) to the Denjoy-Wolff point in terms of the shape of Ω. In particular we prove that the convergence is non-tangential if and only if the domain Ω is “quasi-symmetric with respect to vertical axis”. We also prove that such conditions are equivalent to the curve [0,∞)∋t↦ϕt(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.

Fatou, Julia, and escaping sets in holomorphic (sub)semigroup dynamics

Fatou, Julia, and escaping sets in holomorphic (sub)semigroup dynamics

Diffusion with nonlocal Robin boundary conditions

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.

Remarks on the Operator-Norm Convergence of the Trotter Product Formula

We revise the operator-norm convergence of the Trotter product formula for a pair $$\{A,B\}$$ of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.

Characterization of Log-convex decay in non-selfadjoint dynamics

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.

Diffusion with nonlocal boundary conditions

We consider second order differential operators Aμ on a bounded, Dirichlet regular set Ω⊂Rd, subject to the nonlocal boundary conditionsu(z)=∫Ωu(x)μ(z,dx)for z∈∂Ω. Here the function μ:∂Ω→M+(Ω) is σ(M(Ω),Cb(Ω))-continuous with 0≤μ(z,Ω)≤1 for all z∈∂Ω. Under suitable assumptions on the coefficients in Aμ, we prove that Aμ generates a holomorphic positive contraction semigroup Tμ on L∞(Ω). The semigroup Tμ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in C(Ω‾). We also prove that Tμ is immediately compact and study the asymptotic behavior of Tμ(t) as t→∞.

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.

The Kalton–Lancien theorem revisited: Maximal regularity does not extrapolate

We give a new more explicit proof of a result by Kalton and Lancien stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator A of a holomorphic semigroup which does not have maximal regularity. In particular, we show that there always exists a Schauder basis (fm) such that A can be chosen of the form A(∑m=1∞amfm)=∑m=1∞2mamfm. Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups (Tp(t))t⩾0 on Lp(R) for p∈(1,∞) which have maximal regularity if and only if p=2. These assertions were both open problems. Our approach is completely different than the one of Kalton and Lancien. We use the characterization of maximal regularity by R-sectoriality for our construction.