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Register a new userFractional Sobolev isometric immersions of planar domains
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We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, frac2s}$ regularity for $2/3 le s< 1 $, generalizing the known results on Sobolev and Holder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.
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We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler-Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.
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 $C^{k+2,alpha}$.
We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $sigma=(sigma_1,dots,sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $overline{f} colon {mathbb R}^m to V$ such that $f = sigma circ overline{f}$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1 le p<d/(d-1)$. In the case $m=1$ there always exists a continuous choice $overline{f}$ for given $fcolon {mathbb R} to sigma(V) subseteq {mathbb C}^n$. We give uniform bounds for the $W^{1,p}$-norm of $overline{f}$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $overline{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger Holder class.
Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.
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