Related papers: Characterization of minimizable Lagrangian action …

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.

We consider minimizers $u_\varepsilon$ of the Ginzburg-Landau energy with quadratic divergence penalization on a simply-connected two-dimensional domain $\Omega$. On the boundary, strong tangential anchoring is imposed. We prove that…

Randomized smoothing is a widely adopted technique for optimizing nonsmooth objective functions. However, its efficiency analysis typically relies on global Lipschitz continuity, a condition rarely met in practical applications. To address…

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of…

In this paper we present a cubic regularized Newton's method to minimize a smooth function over a Riemannian manifold. The proposed algorithm is shown to reach a second-order $\epsilon$-stationary point within…

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with…

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to…

In this paper we present a general framework that allows one to study discretization of certain dynamical systems. This generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent bundles and cotangent…

Conservation laws are usually studied in the context of sufficient regularity conditions imposed on the flux function, usually $C^{2}$ and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer…

Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE…

We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our…

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of…

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the langrangian surfaces of a Kaehler surface, or, more generally, a symplectic 4-manifold with suitable…

We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a…

Convergence of the gradient descent algorithm has been attracting renewed interest due to its utility in deep learning applications. Even as multiple variants of gradient descent were proposed, the assumption that the gradient of the…

We establish the well-posedness of viscosity solutions for a class of semi-linear Hamilton-Jacobi equations set on the space of probability measures on the torus. In particular, we focus on equations with both common and idiosyncratic…

The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann…

In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar…

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces…