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Register a new userThe method of moving planes: a quantitative approach
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We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDEs boundary value problems (such as Serrins overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.
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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.
Suppose $uin dot{H}^1(mathbb{R}^n)$. In a seminal work, Struwe proved that if $ugeq 0$ and $|Delta u+u^{frac{n+2}{n-2}}|_{H^{-1}}:=Gamma(u)to 0$ then $dist(u,mathcal{T})to 0$, where $dist(u,mathcal{T})$ denotes the $dot{H}^1(mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwes decomposition with one bubble in all dimensions, namely $delta (u) leq C Gamma (u)$. For Struwes decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $delta(u)leq C Gamma(u)$ when $3leq nleq 5$ while this is false for $ ngeq 6$. In this paper, we show that [dist (u,mathcal{T})leq Cbegin{cases} Gamma(u)left|log Gamma(u)right|^{frac{1}{2}}quad&text{if }n=6, |Gamma(u)|^{frac{n+2}{2(n-2)}}quad&text{if }ngeq 7.end{cases}] Furthermore, we show that this inequality is sharp.
Alexandrovs soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general functions of the principal curvatures. In this short review, we discuss quantitative stability results regarding Alexandrovs theorem which have been obtained by the author in recent years. In particular, we consider hypersurfaces having mean curvature close to a constant and we quantitatively describe the proximity to a single sphere or to a collection of tangent spheres in terms of the oscillation of the mean curvature. Moreover, we also consider the problem in a non local setting, and we show that the non local effect gives a stronger rigidity to the problem and prevents the appearance of bubbling.
We propose a unified method for the large space-time scaling limit of emph{linear} collisional kinetic equations in the whole space. The limit is of emph{fractional} diffusion type for heavy tail equilibria with slow enough decay, and of diffusive type otherwise. The proof is constructive and the fractional/standard diffusion matrix is obtained. The equilibria satisfy a {em generalised} weighted mass condition and can have infinite mass. The method combines energy estimates and quantitative spectral methods to construct a `fluid mode. The method is applied to scattering models (without assuming detailed balance conditions), Fokker-Planck operators and L{e}vy-Fokker-Planck operators. It proves a series of new results, including the fractional diffusive limit for Fokker-Planck operators in any dimension, for which the characterization of the diffusion coefficient was not known, for L{e}vy-Fokker-Planck operators with general equilibria, and in cases where the equilibrium has infinite mass. It also unifies and generalises the results of ten previous papers with a quantitative method, and our estimates on the fluid approximation error seem novel in these cases.
Let $(mathcal{M},g_0)$ be a compact Riemannian manifold-with-boundary. We present a new proof of the classical Gaffneys inequality for differential forms in boundary value spaces over $mathcal{M}$, via the variational approach `{a} la Kozono--Yanagisawa [$L^r$-variational inequality for vector fields and the Helmholtz--Weyl decomposition in bounded domains, Indiana Univ. Math. J. 58 (2009), 1853--1920] combined with global computations based on the Bochners technique.
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