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Konstantinos Zemas

Publications and source records attributed to Konstantinos Zemas.

15 recordsLinked to original sources
arXiv2026

Some lifting and approximation properties for maps in $W^{1,2}(\mathbb{B}^3;\mathbb{S}^2)$

Smooth maps $u\colon\mathbb B^3\to\mathbb S^2$ can be lifted to $\hat u\colon\mathbb B^3\to\mathbb S^3$ using the Hopf fibration $h\colon \mathbb S^3\to\mathbb S^2$ via the factorization $u=h\circ\hat u$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^*ω_{\mathbb S^2}$, and deduce a smooth approximation property preserving the constraint $u^*ω_{\mathbb S^2}=dη$.

math.AP↗
arXiv2026

Atomic Gradient Flows: Gradient Flows on Sparse Representations

One of the most popular approaches for solving total variation-regularized optimization problems in the space of measures are Particle Gradient Flows (PGFs). These restrict the problem to linear combinations of Dirac deltas and then perform a Euclidean gradient flow in the weights and positions, significantly reducing the computational cost while still decreasing the energy. In this work, we generalize PGFs to convex optimization problems in arbitrary Banach spaces, which we call Atomic Gradient Flows (AGFs). To this end, the crucial ingredient turns out to be the right notion of particles, chosen here as the extremal points of the unit ball of the regularizer. This choice is motivated by the Krein-Milman theorem, which ensures that minimizers can be approximated by linear combinations of extremal points. We investigate metric gradient flows of the optimization problem when restricted to such sparse representations, for which we define a suitable discretized functional that we show to be to be consistent with the original problem via the means of $Γ$-convergence. We prove that the resulting evolution of the latter is well-defined using a minimizing movement scheme, and we establish conditions ensuring $λ$-convexity and uniqueness of the flow. Then, using Choquet's theorem, we lift the problem into the Wasserstein space on weights and extremal points, and consider Wasserstein gradient flows in this lifted setting. Our main result is that the lifting of the AGF evolution is again a metric gradient flow in the Wasserstein space, verifying the consistency of the approach with respect to a Wasserstein-type dynamic. Finally, we illustrate the applicability of AGFs to several relevant infinite-dimensional problems, including optimization of functions of bounded variation and curves of measures regularized by Optimal Transport-type penalties.

math.OC↗
arXiv2025

Derivation of Kirchhoff-type plate theories for elastic materials with voids

We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of Santili and Schmidt (2022), by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.

math.AP↗
arXiv2024

On the existence of degenerate solutions of the two-dimensional $H$-system

We consider entire solutions $ω\in\dot H^1(\mathbb R^2;\mathbb R^3)$ of the $H$-system $Δω=2ω_x\wedgeω_y,$ which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.

math.AP↗
arXiv2024

Energy barriers for boundary nucleation in a two-well model without gauge invariance

We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation \cite{knupfer2011minimal} for all rank-1 directions, \textit{but} the ones normal to the confining hyperplane, where the scaling is as in a three-well problem in full space, resulting in a lower energy barrier \cite{Tribuzio-Rueland_1}.

math.AP↗
arXiv2024

Optimal quantitative stability of the Möbius group of the sphere in all dimensions

In any dimension $n\geq 3$, we prove an optimal stability estimate for the Möbius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{λ>0,ϕ\in \mathrm{Möb}(\mathbb S^{n-1})} \int_{\mathbb S^{n-1}}\left|\frac 1λ\nabla_{T} u -\nabla_{ T}ϕ\right|^{n-1} d\mathcal H^{n-1} \leq C_n \mathcal E_{n-1}(u).$ Here, $\mathcal E_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u(\mathbb S^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\geq 4$.

math.DG↗
arXiv2023

Derivation of effective theories for thin 3D nonlinearly elastic rods with voids

We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $Γ$-convergence. Hereby, we generalize the results of the purely elastic setting [57] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending-torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a {discontinuity} of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [32].

math.AP↗
arXiv2023

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations

In this paper we study the convergence of integral functionals with $q$-growth in a randomly perforated domain of $\mathbb R^n$, with $1<q<n$. Under the assumption that the perforations are small balls whose centres and radii are generated by a \emph{stationary short-range marked point process}, we obtain in the critical-scaling limit an averaged analogue of the nonlinear capacitary term obtained by Ansini and Braides in the deterministic periodic case \cite{Ansini-Braides}. In analogy to the random setting introduced by Giunti, Höfer, and Velázquez \cite{Giunti-Hofer-Velasquez} to study the Poisson equation, we only require that the random radii have finite $(n-q)$-moment. This assumption on the one hand ensures that the expectation of the nonlinear $q$-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.

math.AP↗
arXiv2023

Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension

In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are Möbius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal. 239(1):219-299, 2021), with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to Möbius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

math.AP↗
arXiv2023

Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

We present a quantitative geometric rigidity estimate in dimensions $d=2,3$ generalizing the celebrated result by Friesecke, James, and Müller to the setting of variable domains. Loosely speaking, we show that for each $y \in H^1(U;\mathbb{R}^d)$ and for each connected component of a smooth open, bounded set $U \subset \mathbb{R}^d$, the $L^2$-distance of $\nabla y$ from a single rotation can be controlled up to a constant by its $L^2$-distance from the group $SO(d)$, with the constant not depending on the precise shape of $U$, but only on an integral curvature functional related to $\partial U$. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set $U$. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation ($GSBD^2$) for sequences of displacements related to deformations with uniformly bounded elastic energy. As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of $Γ$-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.

math.AP↗
arXiv2022

Rigidity estimates for isometric and conformal maps from $\mathbb{S}^{n-1}$ to $\mathbb{R}^n$

We investigate both linear and nonlinear stability aspects of rigid motions (resp. Möbius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric (resp. conformal) deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. We mostly focus on the case $n=3$, where we also explain why the estimates are optimal in their corresponding settings. In the isometric case the estimate holds true also when $n=2$ and generalizes in dimensions $n\geq 4$ as well, if one requires apriori boundedness in a certain higher Sobolev norm. We also obtain linear stability estimates for both cases in all dimensions. These can be regarded as Korn-type inequalities for the combination of the quadratic form associated with the isometric (resp. conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.

math.AP↗
arXiv2022

From atomistic systems to linearized continuum models for elastic materials with voids

We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discrete-to-continuum analysis, by means of a recent geometric rigidity result in variable domains [27] and Γ-convergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in three-dimensional elasticity.

math.AP↗
arXiv2021

Stability estimates for the conformal group of $\mathbb{S}^{n-1}$ in dimension $n\geq 3$

The purpose of this paper is to exhibit a quantitative stability result for the class of Möbius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The main estimate is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation, an average conformal-isoperimetric type of deficit controls the deviation (in an average sense) of the map in question from a particular Möbius map. The optimality of the result together with its link with the geometric rigidity of the special orthogonal group are also discussed.

math.DG↗