Nonlinear PDEs, calculus of variations and applications

Abstracts

Murat Akman (University of Essex)

Abstract The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity.

In this talk, we study a Minkowski problem for certain measure, called $p$-capacitary surface area measure, associated to a compact convex set $E$ with nonempty interior and its $p$-harmonic capacitary function (solution to the $p$-Laplace equation in the complement of $E$). If $\mu p$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel positive measure $\mu$ on $S^{n-1}$, find necessary and sufficient conditions for which there exists a convex body $E$ with $\mu p=\mu$. We will discuss the existence, uniqueness, and regularity of this problem which have deep connections with the Brunn-Minkowski inequality for $p$-capacity and Monge-Ampère equation.

John Ball (Heriot-Watt University and Maxwell Institute for Mathematical Sciences, Edinburgh)

Abstract A nonlinear elasticity model for comparing images is formulated and analyzed, in which optimal transformations between images are sought as minimizers of an integral functional. The existence of minimizers in a suitable class of homeomorphisms between image domains is established under natural hypotheses, and the question of whether for linearly related images the minimization algorithm delivers the linear transformation as the unique minimizer is discussed. This is joint work with Chris Horner.

Mikhail Cherdantsev (Cardiff University)

Abstract We consider homogenisation problem for convolution type integral operators in a periodic high-contrast medium (consisting of stiff and soft components) with the so-called double porosity type scaling between the components' contrast and the period, while the convolution kernel is subject to a diffusive scaling. We show that the limit operator is of two-scale nature, consisting of the macroscopic part coming from the stiff component – an elliptic second order differential operator with constant coefficients, and the microscopic part coming from the soft component, which remains an integral operator. We then analyse the limiting spectrum and its relation to the spectrum of the limit two-scale operator and discuss similarities and differences between our setting and that one of high-contrast elliptic PDEs.

This a joint work with Andrei Piantnitski (Narvik) and Igor Velcic (Zagreb).

Luigi De Pascale (University of Florence)

Abstract In the framework of optimal transport problems in which one aims to minimize the worst transport cost instead of an average cost, I will discuss the necessity and sufficiency of a common optimality condition called cyclical monotonicity. I will then show how this condition (and a weaker version of it) implies that certain transport plans, among which the local minimizers, have a special geometric structure. (From joint works with Camilla Brizzi (TUM, Munich) and Anna Kausamo (Univ. of Firenze).)

Nicolas Dirr (Cardiff University)

Abstract We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing. We present some numerical results comparing the model to a known exact solution.

Federica Dragoni (Cardiff University)

Abstract Semiconcavity and semiconvexity are key regularity properties for functions with many applications in a broad range of mathematical subjects. The notions of semiconcavity and semiconvexity have been adapted to different geometrical contexts, in particular in sub-Riemannian structures such as Carnot groups, where they turn out to be extremely useful for the study of solutions of degenerate PDEs. In this talk I will show that, for a suitable class of Carnot groups, the Carnot-Carathéodory distance is semiconcave, in the sense of the group, in the whole space. I will also give some applications to solutions of non-coercive Hamilton-Jacobi equations. Joint work with Qing Liu and Ye Zhang from OIST (Okinawa, Japan).

Carolin Kreisbeck (KU Eichstätt-Ingolstadt)

Abstract Motivated by new nonlocal models in hyperelasticity, we discuss a class of variational problems with integral functionals depending on nonlocal gradients, precisely, on truncated versions of the Riesz fractional gradient. We address several aspects regarding the existence theory of these problems and their asymptotic behavior. Our analysis relies on suitable translation operators that allow us to switch between the three types of gradients: classical, fractional, and nonlocal. These provide helpful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, the natural convexity notion in the classical calculus of variations, characterises weak lower semicontinuity also in the fractional and nonlocal setting. As a consequence of a general Gamma-convergence statement, we derive relaxation and homogenization results. Analyzing the limiting behavior as the fractional order tends to 1 or the horizon tends to 0 yields localization to a classical model. This is joint work with Javier Cueto (Universidad Autónoma de Madrid) and Hidde Schönberger (KUEI).

Govind Menon (Brown University)

Abstract The rigorous foundations of Brownian motion on Riemannian manifolds were developed in the 1970s. However, our understanding of this problem, in particular the interplay between the underlying metric and the Brownian motion has been considerably enriched by recent applications, in particular deep learning and turbulence.

We will discuss low-regularity constructions of Brownian motions and a new formulation of the isometric embedding problem that is motivated by these applications. These constructions shed new light on Nash’s work from the 1950s.

This is joint work with Dominik Inauen (Leipzig).

Reza Pakzad (University of Toulon)

Abstract We discuss a few recent rigidity results concerning weakly regular surfaces. In particular, we sketch the proof of the fact that a complete surface of $C^{1,\alpha }$ regularity, whose intrinsic Gaussian curvature is nonnegative, is convex, provided $\alpha >2/3$.

Beatrice Pelloni (Heriot-Watt University)

Abstract I will discuss a very surprising dichotomy in the regularity of solution to dispersive PDE in one spatial variables. This dichotomy leads to a physical effect observed as far back as 1835, but not studied mathematically until the late 90’s. During the course of the following 20 years, many linear and nonlinear results have been obtained in the periodic setting, by Oskolkov, Berry, Rodniansky, Erdogan and Tzirakis among others.

I will give an introduction to this phenomenon, and then show the surprising and very nontrivial effect of generalising one or more of the parameters of the PDE or of the boundary conditions. This is joint work with several collaborators (Boulton, Farmakis, Olver and Smith).

Ana Margarida Ribeiro (Universidade NOVA de Lisboa)

Abstract In this talk I will discuss several notions of convexity that arise in the study of $L^{\mathrm{\infty }}$ minimization problems as well as those that arise from the $L^p$ approximation. The knowledge of variational problems in integral form serves as a guideline for the questions raised. We hope the answers obtained can contribute to establishing a solid foundation for the study of $L^{\mathrm{\infty }}$ problems. This is a joint work with E. Zappale.

Jeyabal Sivaloganathan (University of Bath)

Abstract (This is joint work with P. Negron-Marerro (UPRH).) The Lavrentiev phenomenon in one-dimensional variational problems occurs when the infimum of an integral functional on the class of absolutely continuous functions is strictly lower than the infimum on the class of smooth functions. This phenomenon is significant in the Calculus of Variations as it signals a possible loss of regularity in minimisers and, in some cases, it implies a repulsion property: approximating a global minimizer with smooth functions results in approximate energies that diverge. Consequently, standard numerical methods like the finite element method may fail for these problems.

In this talk, we will demonstrate that a generalised repulsion property applies to three-dimensional elasticity problems exhibiting cavitation and propose a novel scheme that circumvents this issue. Our approach utilizes a modification of the Modica-Mortola functional in which the phase function is coupled with the determinant of the deformation gradient in the elastic stored energy functional. We prove that approximations using this method satisfy the lower bound $\mathrm{\Gamma }$-convergence property in multi-dimensional, non-radial cases. Moreover, we establish convergence to the actual cavitating minimizer for a spherical body under radial deformations.

Florian Theil (University of Warwick)

Abstract The justification of kinetic equations like the Boltzmann equation as a scaling limit of particle dynamics for large times is a longstanding problem; the main challenge is to control the size of the relevant density functions. We propose a regularisation strategy where the number of interactions per particles is bounded and derive uniform-in-time bounds for the discrepancy between the solution of the kinetic equation and the particle dynamics. This result is a consequence of a careful study of cancellations associated with recollisions.

Igor Velcic (University of Zagreb)

Abstract By performing simultaneous homogenization and dimension reduction we discuss different models of elasto-plastic plate, depending on the relation between the thickness of the plate and the period of the oscillations of the material. This is a joint work with Marin Bužančić (University of Zagreb) and Elisa Davoli (University of Vienna).

Arghir Dani Zarnescu (Basque Center for Applied Mathematics (BCAM) and Simion Stoilow Institute of the Romanian Academy)

Abstract We consider a construction proposed by A. Acharya in QAM 2023, LXXXI(1) that builds on the notion of weak solutions for incompressible fluids to provide a scheme that generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them to recover standard solutions. The scheme provides a generalisation of a construction of Y. Brenier for Euler. We rigorously analyze the scheme, extending the work of Brenier for Euler, and also provide an extension of it to the case of Navier-Stokes equations. This is joint work with A. Acharya and B. Stroffolini.

Kewei Zhang (University of Nottingham)

Abstract In this talk I will discuss a smoothing technique for the maximum function in ${\mathbb{R}}^n$, based on the compensated convex transforms to construct some computable multiwell non-negative quasiconvex functions in the vectorial calculus of variations. Let $K\subseteq E\subseteq M^{m\times n}$ with $K$ a finite set in a linear subspace $E\subset M^{m\times n}$ without rank-one matrices, where $M^{m\times n}$ is the space of real $m\times n$ matrices . Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells:

i) Let $f:K\subseteq E\to E_{\perp }$ be an $L$-Lipschitz mapping. We construct computable quasiconvex functions of quadratic growth with finitely many local wells on the graph of $f$ over $K$.

ii) We also construct quasiconvex functions who are of linear growth on $E$ with finitely many local minimizers in $K$, and is of quadratic growth in $E$.

Under certain conditions of our control parameters, our quasiconvex lower bounds are tight in the sense that near each well the lower bound agrees with the original function, and our quasiconvex lower bounds are of $C^{1,1}$. We also consider generalisations of our costructions and other multiwell problems and discuss the implications of our constructions and the corresponding variational problems. This is a joint work with Ke Yin.