arXiv subjects

Michael Hinz

Publications and source records attributed to Michael Hinz.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the existence of optimal shapes in architecture

We consider shape optimization problems for elasticity systems in architecture. A typical question in this context is to identify a structure of maximal stability close to an initially proposed one. We show the existence of such an optimally shaped structure within classes of bounded Lipschitz domains and within wider classes of bounded uniform domains with boundaries that may be fractal. In the first case the optimal shape realizes the infimum of a given energy functional over the class, in the second case it realizes the minimum. As a concrete application we discuss the existence of maximally stable roof structures under snow loads.

math.AP↗

arXiv2020

Approximation of partial differential equations on compact resistance spaces

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

math.FA↗

arXiv2018

Magnetostatic problems in fractal domains

We consider a magnetostatic problem in a 3D "cylindrical" domain of Koch type. We prove existence and uniqueness results for both the fractal and pre-fractal problems and we investigate the convergence of the pre-fractal solutions to the limit fractal one. We consider the numerical approximation of the pre-fractal problems via FEM and we prove a priori error estimates. Some numerical simulations are also shown. Our long term motivation includes studying problems that appear in quantum physics in fractal domains.

math-ph↗

arXiv2018

Canonical diffusions on the pattern spaces of aperiodic Delone sets

We consider pattern spaces of aperiodic and repetitive Delone sets of finite local complexity. These spaces are compact metric spaces and constitute a special class of foliated spaces. We define new Sobolev spaces with respect to the unique invariant measure and prove the existence of the unitary Schrödinger semigroup, which in physics terms describe the evolution of phasons. We define and study natural leafwise diffusion processes on these pattern spaces. These processes have Feller, but lack strong Feller and hypercontractivity properties, and heat kernels do not exist. The associated Dirichlet forms are regular, strongly local, irreducible and recurrent, but not strictly local. For harmonic functions we prove new Liouville and Helmholtz-Hodge type theorems.

math.DS↗

arXiv2018

Sobolev spaces and calculus of variations on fractals

The present note contains a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained mimimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals or on products of fractals.

math.AP↗

arXiv2018

Capacities, removable sets and $L^p$-uniqueness on Wiener spaces

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set $Σ$ of zero Gaussian measure. To prove the equivalence we show the $W^{r,p}(B,μ)$-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We also give connections to Gaussian Hausdorff measures. Roughly speaking, if $L^p$-uniqueness holds then the 'removed' set $Σ$ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least $2p$.

math.FA↗

arXiv2018

A note on Neumann problems on graphs

We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.

math.AP↗

arXiv2017

On the viscous Burgers equation on metric graphs and fractals

We study a formulation of Burgers equation on the Sierpinski gasket, which is the prototype of a p.c.f. self-similar fractal. One possibility is to implement Burgers equation as a semilinear heat equation associated with the Laplacian for scalar functions, just as on the unit interval. Here we propose a second, different formulation which follows from the Cole-Hopf transform and is associated with the Laplacian for vector fields. The difference between these two equations can be understood in terms of different vertex conditions for Laplacians on metric graphs. For the second formulation we show existence and uniqueness of solutions and verify the continuous dependence on the initial condition. We also prove that solutions on the Sierpinski gasket can be approximated in a weak sense by solutions to corresponding equations on approximating metric graphs. These results are part of a larger program discussing nonlinear partial differential equations on fractal spaces.

math.AP↗

arXiv2017

From non-symmetric particle systems to non-linear PDEs on fractals

We present new results and challenges in obtaining hydrodynamic limits for non-symmetric (weakly asymmetric) particle systems (exclusion processes on pre-fractal graphs) converging to a non-linear heat equation. We discuss a joint density-current law of large numbers and a corresponding large deviations principle.

math-ph↗

arXiv2017

Probabilistic characterizations of essential self-adjointness and removability of singularities

We consider the Laplacian and its fractional powers of order less than one on the complement $\mathbb{R}^d\setminusΣ$ of a given compact set $Σ\subset \mathbb{R}^d$ of zero Lebesgue measure. Depending on the size of $Σ$, the operator under consideration, equipped with the smooth compactly supported functions on $\mathbb{R}^d \setminus Σ$, may or may not be essentially self-ajoint. We survey well known descriptions for the critical size of $Σ$ in terms of capacities and Hausdorff measures. In addition, we collect some known results for certain two-parameter stochastic processes. What we finally want to point out is, that, although a priori essential self-adjointness is not a notion directly related to classical probability, it admits a characterization via Kakutani type theorems for such processes.

math.PR↗

arXiv2016

Densely defined non-closable curl on carpet-like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking $1$-forms into $2$-forms cannot be closable if the martingale dimension is larger than one.

math.FA↗

arXiv2016

Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals

We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion harmonicity for finite energy 1-forms. As as corollary, under a certain capacity-separation assumption, we prove that the space of harmonic 1-forms is nontrivial if and only if the classical Čech cohomology is nontrivial. In the examples of classical self-similar fractals these spaces typically are either trivial or infinitely dimensional. Finally, we study Navier-Stokes type models and prove that under our assumptions they have only steady state divergence-free solutions. In particular, we solve the existence and uniqueness problem for the Navier-Stokes and Euler equations for a large class of fractals that are topologically one-dimensional but can have arbitrary Hausdorff and spectral dimensions.

math.FA↗

arXiv2016

Credit default prediction and parabolic potential theory

We consider an approach to credit risk in which the information about the time of bankruptcy is modelled using a Brownian bridge that starts at zero and is conditioned to equal zero when the default occurs. This raises the question whether the default can be foreseen by observing the evolution of the bridge process. Unlike in most standard models for credit risk, we allow the distribution of the default time to be singular. Using a well known fact from parabolic potential theory, we provide a sufficient condition for its predictability.

math.PR↗

arXiv2016

Magnetic fields on resistance spaces

On a metric measure space $X$ that supports a regular, strongly local resistance form we consider a magnetic energy form that corresponds to the magnetic Laplacian for a particle confined to $X$. We provide sufficient conditions for closability and self-adjointness in terms of geometric conditions on the reference measure without assuming energy dominance.

math-ph↗

arXiv2016

Fractal snowflake domain diffusion with boundary and interior drifts

We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector analysis for local Dirichlet forms and PDE on fractals, coercive closed forms, and the analysis of Lipschitz functions.

math.AP↗

arXiv2015

Closability, regularity, and approximation by graphs for separable bilinear forms

We consider a countably generated and uniformly closed algebra of bounded functions. We assume that there is a lower semicontinuous, with respect to the supremum norm, quadratic form and that normal contractions operate in a certain sense. Then we prove that a subspace of the effective domain of the quadratic form is naturally isomorphic to a core of a regular Dirichlet form on a locally compact separable metric space. We also show that any Dirichlet form on a countably generated measure space can be approximated by essentially discrete Dirichlet forms, i.e. energy forms on finite weighted graphs, in the sense of Mosco convergence, i.e. strong resolvent convergence.

math.FA↗

arXiv2015

Magnetic energies and Feynman-Kac-Itô formulas for symmetric Markov processes

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman-Kac-Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.

math.PR↗

arXiv2015

Finite energy coordinates and vector analysis on fractals

We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.

math.PR↗