Abstract We prove the existence and regularity of minimisers for the Canham–Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the 2-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to proving the main results we establish the lower semicontinuity of the Canham–Helfrich energy

Abstract The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These deformations are homeomorphisms \(\, h :{\mathbb {X}} \xrightarrow []{{}_{\!\!\text{ onto }\,\,\!\!}}{\mathbb {Y}}\,\) between domains \(\,{\mathbb {X}}, {\mathbb {Y}} \subset {\mathbb {R}}^n\,\) of the Sobolev class \(\,{\mathscr {W}}^{1,n}_{\text{ loc }} ({\mathbb {X}}

Abstract We propose results that relate the following two contexts: Given a Riemann metric G on \(\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2})\), we find the infimum of the averaged pointwise deficit of an immersion from attaining the orientation-preserving isometric immersion of \(G_{\mid \Omega ^h}\) on \(\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})\), over all weakly regular immersions

Abstract We consider the problem of finding a surface \(\Sigma \subset {\mathbb {R}}^m\) of least Willmore energy among all immersed surfaces having the same boundary, boundary Gauss map and area. Such a problem was considered by S. Germain and S.D. Poisson in the early XIX century as a model for equilibria of thin, clamped elastic plates. We present a solution in the case of boundary data of class

Abstract Continuing the program initiated in Golovaty et al. (SIAM J Math Anal 51(1):276–320, 2018), we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain \(\Omega \) we investigate the \(\varepsilon \rightarrow

Abstract This paper is concerned with two dual aspects of the regularity question for the Navier–Stokes equations. First, we prove a local-in-time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space \(L^3\), then the solution is locally smooth in space for some short time, which is quantified. This

Abstract We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Hölder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications

Abstract We analyse integral representation and \(\varGamma \)-convergence properties of functionals defined on piecewise rigid functions, that is, functions which are piecewise affine on a Caccioppoli partition where the derivative in each component is constant and lies in a set without rank-one connections. Such functionals account for interfacial energies in the variational modeling of materials

Abstract We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the

Abstract We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on \(\mathbb {T}^3 \times \mathbb {R}^3\). Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a

Abstract The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states

Abstract A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the

Abstract We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schrödinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes

Abstract We study the exact controllability for spatially periodic water waves with surface tension, by localized exterior pressures applied to free surfaces. We prove that in any dimension, the exact controllability holds within an arbitrarily short time, for sufficiently small and regular data, provided that the region of control satisfies the geometric control condition. This result was previously

Abstract We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of \(\Gamma \)-development we recover Landau-de Gennes corrections

Abstract We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The exponential stability result proposed by Liu and Rao (Z Angew Math Phys (ZAMP) 57:419–432, 2006) or Tebou (C R Acad Sci Paris Ser I 350: 603–608, 2012) for that system assumes that the damping is localized in a neighborhood of the whole or a part of the boundary under some consideration. In this paper we propose

Abstract Federer’s characterization states that a set \(E\subset \mathbb {R}^n\) is of finite perimeter if and only if \(\mathcal H^{n-1}(\partial ^*E)<\infty \). Here the measure-theoretic boundary \(\partial ^*E\) consists of those points where both E and its complement have positive upper density. We show that the characterization remains true if \(\partial ^*E\) is replaced by a smaller boundary

Abstract Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary

Abstract In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must

Abstract We consider the question raised by Enciso and Peralta-Salas (Arch Ration Mech Anal 220(1):243–260, 2016): what nonconstant functions f can occur as the proportionality factor for a Beltrami field \({{\mathbf {u}}}\) on an open subset \(U \subset \mathbb {R}^3\)? We also consider the related question: for any such f, how large is the space of associated Beltrami fields? By applying Cartan’s

Biofilms are formed when free-floating bacteria attach to a surface and secrete polysaccharide to form an extracellular polymeric matrix (EPS). A general model of biofilm growth needs to include the bacteria, the EPS, and the solvent within the biofilm region Ω(t), and the solvent in the surrounding region D(t). The interface between the two regions, Γ(t), is a free boundary. In this paper, we consider

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of

Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit

The large-time asymptotics of weak solutions to Maxwell-Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential

Minkowski space is shown to be globally stable as a solution to the massive Einstein-Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used