arXiv subjects
Publications and source records attributed to Luc Nguyen.
We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $σ_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.
For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theorem.
We study the problem of prescribing $σ_k$-curvature for a conformal metric on the standard sphere $\mathbb{S}^n$ with $2 \leq k < n/2$ and $n \geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $β\in [2,n)$. We prove among other things the following statements. (1) When $β>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ β= n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $\frac{2}{n-2k}\le β< n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $β< \frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $β$, there exists also some $K$ for which the solution set is empty.
We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,γ}_{\rm loc}$ for any $γ> \frac{1}{k}$. Optimal regularity for solutions to the $σ_k$-Yamabe problem on annuli with finite constant boundary values is also established.
We present results supporting the Horowitz-Myers conjecture, that the Horowitz-Myers metrics minimise energy in the relevant classes of metrics.
Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.
We consider a variational two-dimensional Landau-de Gennes model in the theory of nematic liquid crystals in a disk of radius $R$. We prove that under a symmetric boundary condition carrying a topological defect of degree $\frac{k}{2}$ for some given {\bf even} non-zero integer $k$, there are exactly two minimizers for all large enough $R$. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.
The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized to hyperfine spin states with a Rabi term coupling. This leads to domain wall solutions having a monotone structure for a non-cooperative system. We use the moving plane method to prove mono-tonicity and one-dimensionality of the phase transition solutions. This relies on totally new estimates for a type of system for which no Maximum Principle a priori holds. We also derive that one dimensional solutions are unique up to translations. When the Rabi coefficient is large, we prove that no non-constant solutions can exist.
We prove positivity of energy for a class of asymptotically locally hyperbolic manifolds in dimensions $4\le n \le 7$. The result is established by first proving deformation-of-mass-aspect theorems in dimensions $n\ge 4$. Our positivity results extend to the case $n = 3$ when more stringent conditions are imposed.
We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.
For $ε>0$, 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 every $ε>0$, there exists a unique global minimizer $u_ε$ of this problem; moreover, $u_ε$ is symmetric and of the form $u_ε(x)=f_ε(|x|)\frac{x}{|x|}$ for $x\in B^N$.
We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for $\mathbb{R}^n$-valued maps under a suitable convexity assumption on the potential and for $H^{1/2} \cap L^\infty$ boundary data that is non-negative in a fixed direction $e\in \mathbb{S}^{n-1}$. Furthermore, we show that, when minimisers are not unique, the set of minimisers is generated from any of its elements using appropriate orthogonal transformations of $\mathbb{R}^n$. We also prove corresponding results for harmonic maps
We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partialΩ$ is a smooth bounded hypersurface (of codimension one). When $\partialΩ$ contains a compact smooth submanifold $Σ$ of higher codimension with $\partialΩ\setminusΣ$ being compact, we also give a `sharp' condition for the divergence to infinity of the conformal factor near $Σ$ in terms of the codimension.
We prove that Maxwell fields of asymptotically flat solutions of the Einstein-Maxwell equations inherit the stationarity of the metric.
We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $\nabla^2 ψ+ L(x,ψ,\nabla ψ)$ which are non-decreasing in $ψ$.
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.
We study $k$-radially symmetric solutions corresponding to topological defects of charge $\frac{k}{2}$ for integer $k \neq 0$ in the Landau-de Gennes model describing liquid crystals in two-dimensional domains. We show that the solutions whose radial profiles satisfy a natural sign invariance are stable when $|k| = 1$ (unlike the case $|k|>1$ which we treated before). The proof crucially uses the monotonicity of the suitable components, obtained by making use of the cooperative character of the system. A uniqueness result for the radial profiles is also established.
In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems.