Related papers: Uniform profile near the point defect of Landau-de…
We devise a geometric description of bounded systems at criticality in any dimension $d$. This is achieved by altering the flat metric with a space dependent scale factor $\gamma(x)$, $x$ belonging to a general bounded domain $\Omega$.…
We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is…
We study interior $\varepsilon$-regularity and Type I blowup criteria for suitable weak solutions to the three-dimensional incompressible MHD equations. Our starting point is a direct iteration scheme for the classical…
We investigate the vanishing elasticity limit for minimizers of the Landau-de Gennes model with finite energy. By adopting a refined blow-up and covering analysis, we establish the optimal $ L^p $ ($ 1<p<+\infty $) convergence of minimizers…
We consider the Landau-de Gennes variational model for nematic liquid crystals, in three-dimensional domains. More precisely, we study the asymptotic behaviour of minimizers as the elastic constant tends to zero, under the assumption that…
For $\epsilon>0$, we consider the Ginzburg-Landau functional for $\mathbb R^N$-valued maps defined in the unit ball $B^N\subset \mathbb R^N$ with the vortex boundary data $x$ on $\partial B^N$. In dimensions $N\geq 7$, we prove that for…
We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain $\Omega$ with non-contractible boundary data. Here the tensorial field represents the second moment of a local orientational…
We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of $\mathbb{S}^1$-equivariant (i.e., axially symmetric)…
The aim of this article is twofold. First, in the large-body limit and when the temperature is below the nematic-isotropic transition threshold, we verify that the $\mathbb{S}^2$-valued energy-minimizing harmonic map on a bounded smooth…
We study homogenization of a boundary obstacle problem on $ C^{1,\alpha} $ domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $\gamma$. For any $ \epsilon\in\mathbb{R}_+$, $\partial D=\Gamma \cup \Sigma$,…
In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order~$s \in (0,1/2)$ on a bounded domain $\Omega$. Such a Plateau problem of order $s$ can be reinterpreted as a…
The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the…
We investigate local minimizers of Ginzburg--Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized…
Using $\Gamma$-convergence, we study the Cahn-Hilliard problem with interface width parameter $\varepsilon > 0$ for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional…
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times \Omega:$ \begin{eqnarray*} \partial_t u = \Delta u +\displaystyle \frac{\lambda }{ (1-u)^2 \left( 1…
It was recently found that the classical 3d O$(N)$ model in the semi-infinite geometry can exhibit an "extraordinary-log" boundary universality class, where the spin-spin correlation function on the boundary falls off as $\langle \vec{S}(x)…
We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space $\mathbb{R}^2$ and on the upper half-plane $\mathbb{R}^2_+$, allowing infinite energy. In each case, we…
In this paper, we study the following Lane-Emden system with nearly critical non-power nonlinearity \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{lll} -\Delta u =\frac{|v|^{p-1}v}{[\ln(e+|v|)]^\epsilon}\ \ &{\rm in}\ \Omega,…
We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that…
Let $\Omega \subset \mathbb{R}^n$, for $n \geq 2$, be a bounded $C^2$ domain. Let $q \in L^1_{loc} (\Omega)$ with $q \geq 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for…