Related papers: Loss of double-integral character during relaxatio…

The paper explores various special functions which generalize the two-parametric Mittag-Leffler type function of two variables. Integral representations for these functions in different domains of variation of arguments for certain values…

We prove an integral-representation result for limits of non-local quadratic forms on $H^1_0(\Omega)$, with $\Omega$ a bounded open subset of $\mathbb R^d$, extending the representation on $C^\infty_c(\Omega)$ given by the Beurling-Deny…

Recently, Ali et al defined the class $\mathcal{W}_{\beta}(\alpha, \gamma)$ consisting of functions $f$ which satisfy $$\Re e^{i\phi}\left((1-\alpha+2\gamma)\frac{f(z)}{z}+(\alpha-2\gamma)f'(z)+\gamma zf''(z)-\beta\right)>0,$$ for all $z\in…

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in 0+1 and in 1+1 dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on…

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied with taking into account the minimization of the corresponding integral functional. Such integral, as being not sequentially weakly…

Let $u$ be the first Dirichlet Laplacian eigenfunction of a bounded convex set $\Omega$ in $\mathbb{R}^n$. We strengthen the classical result by Brascamp-Lieb which asserts that $u$ is logconcave in $\Omega$: we prove that, if $u$ is…

We present a method to deform (generically non-abelian) T duals of two-dimensional $\sigma$ models, which preserves classical integrability. The deformed models are identified by a linear operator $\omega$ on the dualised subalgebra, which…

We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N},…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

A measure representation result for a functional modelling optimal design problems for plastic deformations, under linear growth conditions, is obtained. Departing from an energy with a bulk term depending on the deformation gradient and…

In this article we study the quasi-linear equation \[\mathrm{div}\, \mathcal A(x,u,\nabla u)=\mathcal B(x,u,\nabla u)\quad \text{in }\Omega,\qquad u\in H^{1,p}_{loc}(\Omega;w_1dx)\] where $\mathcal A$ and $\mathcal B$ are functions…

Let $G$ be a locally compact group and $\omega$ be a continuous weight on $G$. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$-algebras. As a result we extend previous results of…

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d $\sigma$-models. We focus on the "$\lambda$-model," an integrable model…

We discuss the $\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(\Omega)}^2+\int_\Omega W(u)dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(\Omega)}$ denotes the total contribution from $\Omega$ in the…

We prove that the doubly lambda-deformed sigma-models, which include integrable cases, are canonically equivalent to the sum of two single lambda-deformed models. This explains the equality of the exact beta-functions and current anomalous…

When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon;…

In this paper, we present a new extension of the famous Serrin's lower semicontinuity theorem for the variational functional $\int_{\Omega}f(x,u,u')dx$,we prove its lower semicontinuity in $W_{loc}^{1,1}(\Omega)$ with respect to the strong…

This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that…

We study functionals \begin{equation*} F_\varepsilon (u) := \lambda_\varepsilon \int_\Omega W(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\|\cdot\|_{H^{1/2}}$ when…

We discuss the main features of quantum integrable models taking the classes of universality of the Ising model and the repulsive Lieb-Liniger model as paradigmatic examples. We address the breaking of integrability by means of two…