ترغب بنشر مسار تعليمي؟ اضغط هنا

Non-Convex Planar Harmonic Maps

63   0   0.0 ( 0 )
 نشر من قبل Shahar Kovalsky
 تاريخ النشر 2020
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rado-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tuttes embedding theorem for planar graphs - RKCs discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.



قيم البحث

اقرأ أيضاً

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in $RR^n$.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Holder continuous. In [3 9], F. H. Lin proposed a challenge problem: Can the Holder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.
131 - Nicolas Ginoux 2018
We discuss a method to construct Dirac-harmonic maps developed by J.~Jost, X.~Mo and M.~Zhu in J.~Jost, X.~Mo, M.~Zhu, emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. textbf{59} (2009), no. 11, 1512--1527.The method uses harm onic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension $1$ with target spaces of constant sectional curvature.Before the present article, it remained unclear when the conditions of the theorems in J.~Jost, X.~Mo, M.~Zhu, emph{Some explicit constructions of Dirac-harmonic maps}, J. Geom. Phys. textbf{59} (2009), no. 11, 1512--1527, were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions.In several cases we show the existence of solutions.
In this paper, we will show the Yaus gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $kappa$, $kappageq0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theor ems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.
57 - J. Xiao 2018
For $pin (1,2]$ and a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ let $mu_p(bar{Omega},cdot)$ be the $p$-capacitary curvature measure (generated by the closure $bar{Omega}$ of $Omega$) on the unit circle $mathbb S^1$. This paper shows that such a problem of prescribing $mu_p$ on a planar convex domain: Given a finite, nonnegative, Borel measure $mu$ on $mathbb S^1$, find a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ such that $dmu_p(bar{Omega},cdot)=dmu(cdot)$ is solvable if and only if $mu$ has centroid at the origin and its support $mathrm{supp}(mu)$ does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if $dmu_p(bar{Omega},cdot)=psi(cdot),dell(cdot)$ with $psiin C^{k,alpha}$ and $dell$ being the standard arc-length element on $mathbb S^1$, then $partialOmega$ is of $C^{k+2,alpha}$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا