Related papers: Convergence of regularized nonlocal interaction en…

We investigate the existence of ground states for a free energy functional on Cartan-Hadamard manifolds. The energy, which consists of an entropy and an interaction term, is associated to a macroscopic aggregation model that includes…

We use the method of $\Gamma$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach…

We study a class of variational transmission problems driven by nonlinear energies with discontinuous coefficients across a prescribed interface. The model setting consists of integral functionals of the form \[…

We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$…

We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing…

We consider the Abelian Yang-Mills-Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension $n\geq 3$. This functional is the natural generalisation of the Ginzburg-Landau model…

We investigate local minimizers of Ginzburg--Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized…

We study functionals \begin{equation*} F_\varepsilon (u,\rho) := \frac{1}{\varepsilon} \int_\Omega W(u) \, dx + \frac{1}{|\ln(\varepsilon)|} \int_\Omega \int_\Omega \frac{(u(y) - u(x))^2}{|y - x|^{N+1}} \, dy \,dx +…

In their paper, Braides, Buttazzo and Fragala proved the density of Riemannian energies in the class of Finsler energy functionals with respect to $\Gamma$-convergence in the one-dimensional case. In this thesis we prove that one of the…

This paper is on $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the…

We prove smoothness of $W^{2,2}$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We then derive the $\Gamma$-limit of three dimensional nonlinear shells with inhomogeneous energy…

We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with…

We consider the maximum mean discrepancy ($\mathrm{MMD}$) GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized $\mathrm{MMD}$ GAN. We show that this flow provides a descent…

We consider $\mathbb{S}^2$-valued maps on a domain $\Omega\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $\Omega$ and horizontal penalization on $\partial\Omega$. We first show the…

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…

Flow Matching (FM) has emerged as a powerful paradigm for continuous normalizing flows, yet standard FM implicitly performs an unweighted $L^2$ regression over the entire ambient space. In high dimensions, this leads to a fundamental…

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review…

We use the mathematical framework of loop quantum gravity (LQG) to study the quantization of three dimensional (Riemannian) gravity with positive cosmological constant (Lambda>0). We show that the usual regularization techniques (successful…

We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$…

We use variational convergence to derive a hierarchy of one-dimensional rod theories, starting out from three-dimensional models in nonlinear elasticity subject to local volume-preservation. The densities of the resulting $\Gamma$-limits…