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We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is…
This paper studies the properties of d-stationary points of the trimmed lasso (Luo et al., 2013, Huang et al., 2015, and Gotoh et al., 2018) and the composite optimization problem with the truncated nuclear norm (Gao and Sun, 2010, and…
The performance of penalized likelihood approaches depends profoundly on the selection of the tuning parameter; however, there is no commonly agreed-upon criterion for choosing the tuning parameter. Moreover, penalized likelihood estimation…
A unified classification framework for models of extended plasticity is presented. The models include well known micromorphic and strain gradient plasticity formulations. A unified treatment is possible due to the representation of strain…
This paper is concerned with the problem of representing and learning a linear transformation using a linear neural network. In recent years, there has been a growing interest in the study of such networks in part due to the successes of…
We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on $\mathbb{R}^d$ using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in…
We consider perimeter perturbations of a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. We prove that there exists curves in the…
We consider spatially discrete bistable reaction-diffusion equations that admit wave front solutions. Depending on the parameters involved, such wave fronts appear to be pinned or to glide at a certain speed. We study the transition of…
We consider a beam model representing the transverse deflections of a one dimensional elastic structure immersed in an axial fluid flow. The model includes a nonlinear elastic restoring force, with damping and non-conservative terms…
We present a comprehensive analytical linear stability analysis of the Toner-Tu model for polar active fluids in the ordered phase. Our results provide exact instability criteria and demonstrate that all generic hydrodynamic instabilities…
In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using…
Singularities of even smooth functions are studied. A classification of singular points which appear in typical parametric families of even functions with at most five parameters is given. Bifurcations of singular points near a caustic…
We prove that if a curve parametrized by arc length is a stationary point of the Moebius energy introduced by Jun O'Hara, then it is smooth whenever the Moebius energy is finite. Our methods, interestingly, only rely on purely analytical…
We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of Linear Elastic…
We prove the analyticity of smooth critical points for generalized integral Menger curvature energies $\mathrm{intM}^{(p,2)}$, with $p \in (\tfrac 73, \tfrac 83)$, subject to a fixed length constraint. This implies, together with already…
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants.…
Grain boundaries can block slip-band propagation and generate intense local stress and strain fields that influence subsequent deformation and damage initiation in polycrystalline metals. Conventional geometric criteria, such as Schmid…
Given a smoothly bounded non-contractible domain $\Omega\subset \mathbb{R}^2$, we prove the existence of positive critical points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This is done via degree theory, sharp…
A periodic array of atomic sites, described within a tight binding formalism is shown to be capable of trapping electronic states as it grows in size and gets stubbed by an atom or an atomic clusters from a side in a deterministic way. We…
In this paper we propose a model selection approach to fit a regression model using splines with a variable number of knots. We introduce a penalized criterion to estimate the number and the position of the knots where to anchor the splines…