Related papers: Two optimization problems for the Loewner energy
We apply Epstein's construction of hypersurfaces in the hyperbolic disk $\mathbb D$ to prove identities between the Schwarzian action on $\operatorname{PSL}_2(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, the length of the…
We prove that given a finite set $E$ in a bordered Riemann surface $\mathcal{R}$, there is a continuous map $h\colon \overline{\mathcal{R}}\setminus E\to\mathbb{C}^n$ ($n\geq 2$) such that $h|_{\mathcal{R}\setminus E} \colon…
The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both…
In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions with velocity satisfying additional assumptions in fractional Sobolev spaces with respect to the space variables,…
We consider a pair of isoperimetric problems arising in physics. The first concerns a Schr\"odinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $\Gamma$, formally given by $-\Delta-\alpha…
The Lennard-Jones (LJ) Potential Energy Problem is to construct the most stable form of $N$ atoms of a molecule with the minimal LJ potential energy. This problem has a simple mathematical form $f(x) = 4\sum_{i=1}^N \sum_{j=1,j<i}^N…
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble…
In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of…
We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n} $ to almost everywhere immersed, closed submanifolds of a compact Riemannian…
In this paper, we investigate some of the classical restricted three body problems at the nanoscale, such as the circular planar restricted problem for three C60 fullerenes, and a carbon atom and two C60 fullerenes. We model the van der…
This research studies a non-convex geometric optimization problem arising from the field of optical wireless power transfer. In the considered optimization problem, the cost function is a sum of negatively and fractionally powered distances…
We consider energy-critical Schroedinger maps with target either the sphere S^2 or hyperbolic plane H^2 and establish that a unique solution may be continued so long as a certain space-time L^4 norm remains bounded. This reduces the large…
We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an…
The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner M\"obius transformations of R^n. We show that any torus in R^3 with energy at most $8 \pi-delta$ has a representative under…
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and…
This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the…
We study the pure Neumann Lane-Emden problem in a bounded domain \[ -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on }\partial \Omega, \] in the subcritical, critical, and supercritical regimes. We show existence…
This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for…
Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which…
We consider the fundamental task of optimising a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in…