arXiv subjects
Publications and source records attributed to Armin Schikorra.
We consider the scaling-invariant nonlocal Willmore energy, defined via the nonlocal mean curvature by Caffarelli, Roquejoffre and Savin. Our main result is the existence of minimizers in the class of convex $C^1$-curves.
We extend the Calderón-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations $\mathcal{L}^{s}_{A} u = f$, where the operator $\mathcal{L}^{s}_{A}$ is formally given by \[ \mathcal{L}^s_{A}u = \int_{\mathbb{R}^n}\frac{A(x, y)}{\vert x-y\vert ^{n+2s}} \frac{(x-y)\otimes (x-y)}{\vert x-y\vert ^2}(u(x)-u(y))dy. \] For $0 < s < 1$ and $A:\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}$ taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier-Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot, y)$ is uniformly Holder continuous and $\inf_{x\in \mathbb{R}^n}A(x, x) > 0$, then for $f\in L^{p}_{loc},$ for $p\geq 2$, the solution vector $u\in H^{2s-δ,p}_{loc}$ for some $δ\in (0, s)$.
We develop a new method leading to an elementary proof of a generalization of Gromov's theorem about non existence of Hölder embeddings into the Heisenberg group.
We investigate surfaces with bounded L^p-norm of the fractional mean curvature, a quantity we shall refer to as fractional Willmore-type functional. In the subcritical case and under convexity assumptions we show how this Willmore-functional controls local parametrization, and conclude as consequences lower Ahlfors-regularity, a weak Michael-Simon type inequality, and an application to stability.
We show continuity of solutions $u \in W^{1,n}(B^n,\mathbb{R}^N)$ to the system \[ -{\rm div} (|\nabla u|^{n-2} \nabla u) = Ω\cdot |\nabla u|^{n-2} \nabla u \] when $Ω$ is an $L^n$-antisymmetric potential -- and additionally satisfies a Lorentz-space assumption. To obtain our result we study a rotated n-Laplace system \[ -{\rm div} (Q|\nabla u|^{n-2} \nabla u) = \tildeΩ \cdot |\nabla u|^{n-2} \nabla u, \] where $Q \in W^{1,n}(B^n,SO(N))$ is the Coulomb gauge which ensures improved Lorentz-space integrability of $\tildeΩ$. Because of the matrix-term $Q$, this system does not fall directly into Kuusi-Mingione's vectorial potential theory. However, we adapt ideas of their theory together with Iwaniec' stability result to obtain $L^{(n,\infty)}$-estimates of the gradient of a solution which, by an iteration argument leads to the regularity of solutions. As a corollary of our argument we see that $n$-harmonic maps into manifolds are continuous if their gradient belongs to the Lorentz-space $L^{(n,2)}$ -- which is a trivial and optimal assumption if $n=2$, and the weakest assumption to date for the regularity of critical $n$-harmonic maps, without any added differentiability assumption. We also discuss an application to H systems.
Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of $π_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}: π_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to \mathcal{N}$ we give a quantitative estimate of its rational homotopy group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or Hölder norm. That is, we show that for all $β\in (β_0({\rm deg}),1]$, \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{β,\frac{N}β}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}, \] and \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^β(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $β_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.
Extending an argument by Shatah and Struwe we obtain uniqueness for solutions of the half-wave map equation in dimension $d \geq 3$ in the natural energy class.
This paper is part of an undergraduate research project. We discuss the Heisenberg group H1, the three-dimensional space R3 equipped with one of two equivalent metrics, the Koranyi- and Carnot- Caratheodory metric. We show that the notion of length of curves for both metrics coincide, and that shortest curves, so-called geodesics, exist.
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
We discuss tangential Sobolev-estimates up to the boundary for solutions to the regional fractional laplacian on the upper half-plane. These estimates can be used to reduce the boundary Calderon-Zygmund theory of any dimension to a one-dimensional nonlocal problem.
Motivated by problems where jumps across lower dimensional subsets and sharp transitions across interfaces are of interest, this paper studies the properties of fractional bounded variation ($BV$)-type spaces. Two different natural fractional analogs of classical $BV$ are considered: $BV^α$, a space induced from the Riesz-fractional gradient that has been recently studied by Comi-Stefani; and $bv^α$, induced by the Gagliardo-type fractional gradient often used in Dirichlet forms and Peridynamics - this one is naturally related to the Caffarelli-Roquejoffre-Savin fractional perimeter. Our main theoretical result is that the latter $bv^α$ actually corresponds to the Gagliardo-Slobodeckij space $W^{α,1}$. As an application, using the properties of these spaces, novel image denoising models are introduced and their corresponding Fenchel pre-dual formulations are derived. The latter requires density of smooth functions with compact support. We establish this density property for convex domains.
We prove that for antisymmetric vectorfield $Ω$ with small $L^2$-norm there exists a gauge $A \in L^\infty \cap \dot{W}^{1/2,2}(\mathbb{R}^1,GL(N))$ such that ${\rm div}_{\frac12} (AΩ- d_{\frac{1}{2}} A) = 0$. This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.
Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_Ω \int_Ω K\left (x,|x-y|,\frac{x-y}{|x-y|}\right ) \frac{(u(x)-u(y))(φ(x)-φ(y))}{|x-y|^{n+2s}} dx dy = g[φ], \quad \forall ϕ\in C_c^{\infty}(Ω) \] where $Ω\subset \mathbb{R}^{n}$ is an open set. Here we assume $K$ is bounded, nonnegative and continuous in the first entry -- and ellipticity is ensured by assuming that $K$ is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional $p$-Laplace equations around points where $u \in C^1$ and $|\nabla u| \neq 0$.
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 Hölder 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.
This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods.
We investigate minimizers and critical points for scale-invariant tangent-point energies ${\rm TP}^{p,q}$ of closed curves. We show that a) minimizing sequences in ambient isotopy classes converge to locally critical embeddings in all but finitely many points and b) show regularity of locally critical embeddings. Technically, the convergence theory a) is based on a gap-estimate of a fractional Sobolev spaces in comparison to the tangent-point energy. The regularity theory b) is based on constructing a new energy $\mathcal{E}^{p,q}$ and proving that the derivative $γ'$ of a parametrization of a ${\rm TP}^{p,q}$-critical curve $γ$ induces a critical map with respect to $\mathcal{E}^{p,q}$ acting on torus-to-sphere maps.
The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Ω)$ compactly embeds in $W^{-1,q}(Ω)$, for every $q < 2$ and for any open and bounded set $Ω$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities.
We consider minimizing harmonic maps $u$ from $Ω\subset \mathbb{R}^n$ into a closed Riemannian manifold $\mathcal{N}$ and prove: (1) an extension to $n \geq 4$ of Almgren and Lieb's linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$ is finite, we have \[ \mathcal{H}^{n-3}(\textrm{sing } u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d \mathcal{H}^{n-1}; \] (2) an extension of Hardt and Lin's stability theorem. Namely, assuming that the target manifold is $\mathcal{N}=\mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s \in (\frac{1}{2},1]$ and $p \in [2,\infty)$ satisfying $sp \geq 2$. We also discuss sharpness.