Four point functions in CFT’s with slightly broken higher spin symmetry

We compute spinning four point functions in the quasi-fermionic three dimensional conformal field theory with slightly broken higher spin symmetry at finite t’Hooft coupling. More concretely, we obtain a formula for $$ \left\langle {j}_s{j}_{\tilde{0}}{j}_{\tilde{0}}{j}_{\tilde{0}}\right\rangle $$ j s j 0 ˜ j 0 ˜ j 0 ˜ , where js is a higher spin current and $$ {j}_{\tilde{0}} $$ j 0 ˜ is the scalar single trace operator. Our procedure consists in writing a plausible ansatz in Mellin space and using crossing, pseudo-conservation and Regge boundedness to fix all undetermined coefficients. Our method can potentially be generalised to compute all spinning four point functions in these theories.

Two-point correlator of chiral primary operators with a Wilson line defect in $$ \mathcal{N} $$ = 4 SYM

We study the two-point function of the stress-tensor multiplet of $$ \mathcal{N} $$ N = 4 SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the 20′ irrep of the $$ \mathfrak{so} $$ so (6)R R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of $$ \mathcal{N} $$ N = 4 SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques.

Solenoidal extensions in domains with obstacles: explicit bounds and applications to Navier–Stokes equations

We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem with the resulting divergence to obtain a solenoidal extension; finally, by solving a variational problem involving the infinity-Laplacian and using ad hoc cutoff functions, we find explicit bounds in terms of the geometric parameters of the obstacle. The natural applications of our results lie in the analysis of inflow–outflow problems, in which an explicit bound on the inflow velocity is needed to estimate the threshold for uniqueness in the stationary Navier–Stokes equations and, in case of symmetry, the stability of the obstacle immersed in the fluid flow.

On the continuity of the trace operator in GSBV (Ω) and GSBD (Ω)

In this paper, we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed (n − 1)-dimensional set Γ, with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.

Trace Operator Inequality for Superquadratic Functions

In this work, some operator trace inequalities are proved. An extension of Klein's inequality for all Hermitian matrices is proved. A non-commutative version (or Hansen-Pedersen version) of the Jensen trace inequality is provided as well. A generalization of the result for any positive Hilbert space operators acts on a positive unital linear map is established.

A polyconvex extension of the logarithmic Hencky strain energy

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [Formula: see text] which is equal to the classical Hencky strain energy [Formula: see text] in a neighborhood of the identity matrix 𝟙; here, [Formula: see text] denotes the set of [Formula: see text]-matrices with positive determinant, [Formula: see text] denotes the deformation gradient, [Formula: see text] is the corresponding stretch tensor, [Formula: see text] is the principal matrix logarithm of [Formula: see text], [Formula: see text] is the trace operator, [Formula: see text] is the Frobenius matrix norm and [Formula: see text] is the deviatoric part of [Formula: see text]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [Formula: see text] in the so-called Valanis–Landel form [Formula: see text] with [Formula: see text], where [Formula: see text] denote the singular values of [Formula: see text].

Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H1 (Ω)N → H1/2(∂Ω); u ↦ u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(hα + ε) and O(h2α + ε) for the velocity in the H1- and L2-norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.