arXiv subjects

Peter Gladbach

Publications and source records attributed to Peter Gladbach.

16 recordsLinked to original sources

arXiv2024

Stochastic homogenisation of nonlinear minimum-cost flow problems

This paper deals with the large-scale behaviour of nonlinear minimum-cost flow problems on random graphs. In such problems, a random nonlinear cost functional is minimised among all flows (discrete vector-fields) with a prescribed net flux through each vertex. On a stationary random graph embedded in $\mathbb{R}^d$, our main result asserts that these problems converge, in the large-scale limit, to a continuous minimisation problem where an effective cost functional is minimised among all vector fields with prescribed divergence. Our main result is formulated using $\Gamma$-convergence and applies to multi-species problems. The proof employs the blow-up technique by Fonseca and M\"uller in a discrete setting. One of the main challenges overcome is the construction of the homogenised energy density on random graphs without a periodic structure.

math.AP↗

arXiv2024

Stochastic homogenization of dynamical discrete optimal transport

The aim of this paper is to examine the large-scale behavior of dynamical optimal transport on stationary random graphs embedded in $\R^n$. Our primary contribution is a stochastic homogenization result that characterizes the effective behavior of the discrete problems in terms of a continuous optimal transport problem, where the homogenized energy density results from the geometry of the discrete graph.

math.PR↗

arXiv2024

Connecting disclinations by ridges

We consider a thin elastic sheet with a finite number of disclinations in a variational framework in the F\"oppl-von K\'arm\'an approximation. Under the non-physical assumption that the out-of-plane displacement is a convex function, we prove that minimizers display ridges between the disclinations. We prove the associated energy scaling law with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet. One of the key estimates in the proof that we consider of independent interest is a generalization of the monotonicity property of the Monge-Amp\`ere measure.

math.AP↗

arXiv2024

Variational interacting particle systems and Vlasov equations

We consider optimization problems for interacting particle systems. We show that critical points solve a Vlasov equation, and that in general no minimizers exist despite continuity of the action functional. We prove an explicit representation of the relaxation of the action functional. We show convergence of N-particle minimizers to minimizers of the relaxed action, and finally characterize minimizers of dynamic interacting particle optimal transport problems as solutions to Hamilton-Jacobi-Bellman equations.

math.AP↗

arXiv2024

Variational competition between full Hessian and its determinant for convex functions

We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its determinant, where the former is treated as a small perturbation in the space $L^2$ and the latter as the leading-order term, in the negative Sobolev space $W^{-2,2}$. We point out how this setting is motivated by problems in nonlinear elasticity, and obtain a corollary for a variational problem based on the so-called Föppl-von-Kármán energy.

math.AP↗

arXiv2023

The Euler-Bernoulli limit of thin brittle linearized elastic beams

We show that the linear brittle Griffith energy on a thin rectangle $Γ$-converges after rescaling to the linear one-dimensional brittle Euler-Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy

math.AP↗

arXiv2023

Consistent and convergent discretizations of Helfrich-type energies on general meshes

We show that integral curvature energies on surfaces of the type $E_0(M) := \int_M f(x,n_M(x),D n_M(x))\,d\mathcal{H}^2(x)$ have discrete versions for triangular complexes, where the shape operator $D n_M$ is replaced by the piecewise gradient of a piecewise affine edge director field. We combine an ansatz-free asymptotic lower bound for any uniform approximation of a surface with triangular complexes and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director.

math.AP↗

arXiv2023

Non-Newtonian thin-film equations: global existence of solutions, gradient-flow structure and guaranteed lift-off

We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solutions to the power-law thin-film equation \begin{equation*} \partial_t u + \partial_x\bigl(m(u) |\partial_x^3 u - G^{\prime\prime}(u) \partial_x u|^{α-1} \bigl(\partial_x^3 u - G^{\prime\prime}(u) \partial_x u\bigr)\bigr) = 0 \end{equation*} with potential $G$ in the shear-thinning ($α> 1$), Newtonian ($α= 1$) and shear-thickening case ($0 <α< 1$). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case $G\equiv 0$ which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.

math.AP↗

arXiv2022

An anisotropic Poincaré inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals

We show that functions in $GSBV^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $Γ$-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.

math.AP↗

arXiv2021

Homogenisation of dynamical optimal transport on periodic graphs

This paper deals with the large-scale behaviour of dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a $Γ$-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.

math.AP↗

arXiv2021

Approximation of the Willmore energy by a discrete geometry model

We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $Γ$-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.

math.AP↗

arXiv2021

Limits of density-constrained optimal transport

We consider the problem of dynamic optimal transport with a density constraint. We derive variational limits in terms of $Γ$-convergence for two singular phenomena. First, for densities constrained near a hyperplane we recover the optimal flow through an infinitesimal permeable membrane. Second, for rapidly oscillating periodic constraints we obtain the optimal flow through a homogenized porous medium.

math.AP↗

arXiv2020

Scaling limits of discrete optimal transport

We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in $\mathbb{R}^d$. These metrics are natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric $\mathcal{W}_{\mathcal{T}}$ in terms of $\mathbb{W}_2$, as the size of the mesh $\mathcal{T}$ tends to $0$. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.

math.AP↗

arXiv2020

Homogenisation of one-dimensional discrete optimal transport

This paper deals with dynamical optimal transport metrics defined by spatial discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow formulations for dissipative PDE. However, it has recently been shown that these metrics do not in general converge to $W_2$, unless strong geometric constraints are imposed on the discrete mesh. In this paper we prove that, in a $1$-dimensional periodic setting, discrete transport metrics converge to a limiting transport metric with a non-trivial effective mobility. This mobility depends sensitively on the geometry of the mesh and on the non-local mobility at the discrete level. Our result quantifies to what extent discrete transport can make use of microstructure in the mesh to reduce the cost of transport.

math.AP↗

arXiv2019

Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces

We prove weak and strong versions of the coarea formula and the chain rule for distributional Jacobian determinants $Ju$ for functions $u$ in fractional Sobolev spaces $W^{s,p}(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. The weak forms of the formulae are proved for the range $sp>n-1$, $s> \frac{n-1}{n}$, while the strong versions are proved for the range $sp\geq n$, $s\geq \frac{n}{n+1}$. We also provide a chain rule for distributional Jacobian determinants of Hölder functions and point out its relation to two open problems in geometric analysis.

math.AP↗

arXiv2018

Solvation in the Large Box Limit

In this paper, the authors study the limit of a sharp interface model for the solvation of charged molecules in an implicit solvent as the number of solute molecules and the size of the surrounding box tend to infinity. The energy is given by a combination of local terms accounting for the physical presence of the molecules in the solvent and a nonlocal electrical energy with or without an ionic effect. In the presence of an ionic effect, the authors prove a screening effect in the limit, i.e., the limit is completely localized and hence electrical long-range interactions of the molecules can be neglected. In the absence of the ionic effect, the authors show that the behavior of the energy depends on the scaling of the number of molecules with respect to the size of the surrounding box. All scaling regimes are identified and corresponding limit results proved. In regimes with many solute molecules this limit includes electrical interactions of $H^{-1}$-type between the molecules.

math.AP↗