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Register a new userTorus-like solutions for the Landau-de Gennes model. Part II: Topology of $mathbb{S}^1$-equivariant minimizers
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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) configurations. We assume smooth and nonvanishing $mathbb{S}^1$-equivariant (e.g. homeotropic) Dirichlet boundary conditions and a physically relevant norm constraint (Lyuksyutov constraint) in the interior. Relying on results in cite{DMP1} in the nonsymmetric setting, we prove partial regularity of minimizers away from a possible finite set of interior singularities lying on the symmetry axis. For a suitable class of domains and boundary data we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite union of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit $mathbb{S}^1$-equivariant harmonic maps into $mathbb{S}^4$, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.
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We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the boundary for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of boundary data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.
We consider a variant of Gamows liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative extension of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.
In this paper, we study the connection between the Ericksen-Leslie equations and the Beris-Edwards equations in dimension two. It is shown that the weak solutions to the Beris-Edwards equations converge to the one to the Ericksen-Leslie equations as the elastic coefficient tends to zero. Moreover, the limiting weak solutions to the Ericksen-Leslie equations may have singular points.
We construct entire solutions of the magnetic Ginzburg-Landau equations in dimension 4 using Lyapunov-Schmidt reduction. The zero set of these solutions are close to the minimal submanifolds studied by Arezzo-Pacardcite{Arezzo}. We also show the existence of a saddle type solution to the equations, whose zero set consists of two vertical planes in $mathbb{R}^4$. These two types of solutions are believed to be energy minimizers of the corresponding energy functional and lie in the same connect component of the moduli space of entire solutions.
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in cite{Seis16a} and some tools from harmonic analysis introduced in cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in cite{CrippaSpirito15}.
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