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Markus Kunze

Publications and source records attributed to Markus Kunze.

At least 37 records · Page 2Linked to original sources
arXiv2020

The Bi-Laplacian with Wentzell boundary conditions on Lipschitz domains

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $Ω\subseteq\mathbb{R}^d$ with Lipschitz boundary $Γ$. More precisely, using form methods, we show that the associated operator on the ground space $L^2(Ω)\times L^2(Γ)$ has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

math.AP↗
arXiv2019

Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations

We prove convergence of the solutions X_n of semilinear stochastic evolution equations dX_n(t) = (A_nX(t) + F_n(t,X_n(t)))dt + G_n(t,X_n(t))dW_H(t), X_n(0) = x_n, on a Banach space B, driven by a cylindrical Brownian motion W_H in a Hilbert space H. We assume that the operators A_n converge to A and the locally Lipschitz functions F_n and G_n converge to the locally Lipschitz functions F and G in an appropriate sense. Moreover, we obtain estimates for the lifetime of the solution X of the limiting problem in terms of the lifetimes of the approximating solutions X_n. We apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions. The operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator.

math.PR↗
arXiv2018

Irregular convergence of mild solutions of semilinear equations

We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters

math.FA↗
arXiv2018

Comments on the paper 'Static solutions of the Vlasov-Einstein system' by G. Wolansky

In this note we address the attempted proof of the existence of static solutions to the Einstein-Vlasov system as given in \cite{Wol}. We focus on a specific and central part of the proof which concerns a variational problem with an obstacle. We show that two important claims in \cite{Wol} are incorrect and we question the validity of a third claim. We also discuss the variational problem and its difficulties with the aim to stimulate further investigations of this intriguing problem: to answer the question whether or not static solutions of the Einstein-Vlasov system can be found as local minimizers of an energy-Casimir functional.

gr-qc↗
arXiv2017

${L^p}$-theory for Schrödinger systems

In this article we study for $p\in (1,\infty)$ the $L^p$-realization of the vector-valued Schrödinger operator $\mathcal{L}u := \mathrm{div} (Q\nabla u) + V u$. Using a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss, we prove that the $L^p$-realization of $\mathcal{L}$, defined on the intersection of the natural domains of the differential and multiplication operators which form $\mathcal{L}$, generates a strongly continuous contraction semigroup on $L^p(\mathbb{R}^d; \mathbb{R}^m)$. We also study additional properties of the semigroup such as extension to $L^1$, positivity, ultracontractivity and prove that the generator has compact resolvent.

math.AP↗
arXiv2017

Diffusion with nonlocal Robin boundary conditions

We investigate a second order elliptic differential operator $A_{β, μ}$ on a bounded, open set $Ω\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq β\in L^{\infty}(\partialΩ)$ and $μ\colon \partial Ω\to \mathscr{M}(\overlineΩ)$, and boundary conditions of the form \[ \partial_ν^{\mathscr{A}}u(z)+β(z)u(z)=\int_{\overlineΩ}u(x)μ(z)(dx),\ z\in\partialΩ, \] where $\partial_ν^{\mathscr{A}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $μ$ we show that $A_{β, μ}$ generates a holomorphic semigroup $T_{β,μ}$ on $L^{\infty}(Ω)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overlineΩ)$. Its restriction to $C(\overlineΩ)$ 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.

math.FA↗
arXiv2017

An averaging principle for fast diffusions in domains separated by semi-permeable membranes

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed towards the end of the paper.

math.FA↗
arXiv2016

Diffusion with nonlocal boundary conditions

We consider second order differential operators $A_μ$ on a bounded, Dirichlet regular set $Ω\subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_Ωu(x)\, μ(z, dx)\quad \mbox{for } z \in \partial Ω. \] Here the function $μ: \partialΩ\to \mathscr{M}^+(Ω)$ is $σ(\mathscr{M} (Ω), C_b(Ω))$-continuous with $0\leq μ(z,Ω) \leq 1$ for all $z\in \partial Ω$. Under suitable assumptions on the coefficients in $A_μ$, we prove that $A_μ$ generates a holomorphic positive contraction semigroup $T_μ$ on $L^\infty(Ω)$. The semigroup $T_μ$ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in $C(\barΩ)$. We also prove that $T_μ$ is immediately compact and study the asymptotic behavior of $T_μ(t)$ as $t \to \infty$.

math.FA↗
arXiv2014

A Pettis-Type Integral and Applications to Transition Semigroups

Motivated by applications to transition semigroups, we introduce the notion of a norming dual pair and study a Pettis-type integral on such pairs. In particular, we establish a sufficient condition for integrability. We also introduce and study a class of semigroups on such dual pairs which are an abstract version of transition semigroups. Using our results, we give conditions ensuring that a semigroup consisting of kernel operators has a Laplace transform which also consists of kernel operators. We also provide conditions under which a semigroup is uniquely determined by its Laplace transform.

math.FA↗
arXiv2013

On the lattice structure of weakly continuous operators on the space of measures

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.

math.FA↗
arXiv2012

Rotating, stationary, axially symmetric spacetimes with collisionless matter

The existence of stationary solutions to the Einstein-Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat non-vacuum spacetimes. If angular momentum is allowed to be non-zero, the system of equations to solve contains one semilinear elliptic equation which is singular on the axis of rotation. This can be handled very efficiently by recasting the equation as one for an axisymmetric unknown on $\R^5$.

gr-qc↗
arXiv2012

Mean ergodic theorems on norming dual pairs

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.

math.FA↗
arXiv2011

Approximating the coefficients in semilinear stochastic partial differential equations

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt + G(t,X(t))dW_H(t), X(0)=X_0, where W_H is a cylindrical Brownian motion on a Hilbert space H. We prove continuous dependence of the compensated solutions X(t)-e^{tA}X_0 in the norms L^p(Ω;C^λ([0,T];E)) assuming that the approximating operators A_n are uniformly sectorial and converge to A in the strong resolvent sense, and that the approximating nonlinearities F_n and G_n are uniformly Lipschitz continuous in suitable norms and converge to F and G pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite-dimensional multiplicative noise.

math.PR↗
arXiv2010

Existence of axially symmetric static solutions of the Einstein-Vlasov system

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.

gr-qc↗