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Armin Schikorra

Publications and source records attributed to Armin Schikorra.

At least 73 records · Page 4Linked to original sources
arXiv2017

On regularity theory for n/p-harmonic maps into manifolds

In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in \dot{H}^{s,p}(\mathbb{R}^n,\mathcal{N})$ and ${\mathcal{N}}\subset\mathbb{R}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p \leq 2$. For $p > 2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory is in the two cases based on regularity results for nonlocal Schrödinger systems with an antisymmetric potential.

math.AP↗
arXiv2017

Boundary regularity for conformally invariant variational problems with Neumann data

We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support manifold. For example, harmonic maps, or $H$-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Riviere, that these maps satisfy a system with an antisymmetric potential, from which one can derive regularity of the solution. We show that these maps satisfy along the boundary a system with a nonlocal antisymmetric boundary potential which contains information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.

math.AP↗
arXiv2017

On energy-critical half-wave maps into $\mathbb{S}^2$

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $\mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Möbius group on the target $\mathbb{S}^2$ applied to half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $\mathbf{Q}$ with arbitrary degree $m \geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $\mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $\mathbb{R}$ to the unit circle $\mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.

math.AP↗
arXiv2017

Fractional div-curl quantities and applications to nonlocal geometric equations

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to Hélein's for harmonic maps into spheres. Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivière's celebrated argument in the local case. Lastly, the fractional div-curl quantities provide also a new, simpler, proof for Hölder continuity of $W^{s,n/s}$-harmonic maps into spheres and we extend this to an argument for $W^{s,n/s}$-harmonic maps into homogeneous targets. This is an analogue of Strzelecki's and Toro-Wang's proof for $n$-harmonic maps into spheres and homogeneous target manifolds, respectively.

math.AP↗
arXiv2016

Sharp commutator estimates via harmonic extensions

We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da Lio-Rivière three-term commutators. We also give a new limiting $L^1$-estimate for a double commutator of Coifman-Rochberg-Weiss-type, and several intermediate estimates. The beauty of our method is that all those commutator estimates, which are originally proven by various specific methods or by general paraproduct arguments, can be obtained purely from integration by parts and trace theorems. Another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way.

math.AP↗
arXiv2016

An $L^1$-type estimate for Riesz potentials

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $α\in (0, N)$: \[ \|I_αu\|_{L^{N/(N-α)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev inequalities for the Riesz fractional gradient.

math.FA↗
arXiv2016

Regularity for a fractional p-Laplace equation

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian situation, namely $C^{s+α}_{loc}$-regularity for the homogeneous equation.

math.AP↗
arXiv2016

Invitation to H-systems in higher dimensions: known results, new facts, and related open problems

In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions $n\ge 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfaces of prescribed mean curvature) and $n$-harmonic maps into compact Riemannian manifolds. For $n=2$ several solutions of these problems are known but they all break down in higher dimensions (unless one considers special cases, e.g. hypersurfaces of constant mean curvature or manifolds with symmetries). We discuss some of the known proofs and hint at the main difficulties. We also state a few new results (such as positive answers for all solutions of class $W^{n/2,2}$ for even $n$, instead of $W^{1,n}$) and list some open questions of independent interest - including specific endpoint variants of the Coifman-Rochberg-Weiss theorem, addressing the boundedness of commutators of fractional and singular integrals with multiplication by bounded functions of class $W^{1,n}$ - that would lead to solutions of these two problems.

math.AP↗
arXiv2015

Nonlinear commutators for the fractional p-Laplacian and applications

We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weak fractional $p$-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $\frac{n}{s}$-harmonic maps converge strongly outside at most finitely many points.

math.AP↗
arXiv2015

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $Ω\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(Ω)$ of solutions to the non-local equation $$(-Δ)^\frac n2 u_k=λ_k u_ke^{\frac n2 u_k^2}\quad \text{in }Ω,$$ where $λ_k\toλ_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(Ω)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $Ω$ and with $(-Δ)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(Ω)$, a suitably rescaled subsequence $η_k$ converges to the function $η_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-Δ)^\frac n2 η=(n-1)!e^{nη}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivière when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $Λ:=\limsup_{k\to \infty}\|(-Δ)^\frac n4 u_k\|_{L^2}^2\ge Λ_1:= (n-1)!|S^n|$.

math.AP↗
arXiv2015

$L^p$-theory for fractional gradient PDE with VMO coefficients

In this paper, we prove $L^p$ estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are $VMO$. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations.

math.AP↗
arXiv2014

Lp-gradient harmonic maps into spheres and SO(N)

We consider critical points of the energy $E(v) := \int_{\mathbb{R}^n} |\nabla^s v|^{\frac{n}{s}}$, where $v$ maps locally into the sphere or $SO(N)$, and $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ is the formal fractional gradient, i.e. $\partial_α^s$ is a composition of the fractional laplacian with the $α$-th Riesz transform. We show that critical points of this energy are Hölder continuous. As a special case, for $s = 1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere, which is interesting on its own.

math.AP↗
arXiv2014

A notion of nonpositive curvature for general metric spaces

We introduce a new definition of nonpositive curvature in metric spaces and study its relationship to the existing notions of nonpositive curvature in comparison geometry. The main feature of our definition is that it applies to all metric spaces and does not rely on geodesics. Moreover, a scaled and a relaxed version of our definition are appropriate in discrete metric spaces, and are believed to be of interest in geometric data analysis.

math.MG↗
arXiv2014

Integro-differential harmonic maps into spheres

We introduce (integro-differential) harmonic maps into spheres, which are defined as critical points of the Besov-Slobodeckij energy $\int\limits_Ω\int\limits_Ω \frac{|v(x)-v(y)|^{p_s}}{|x-y|^{n+sp_s}}\ dx\ dy$. For $p_s = 2$ these are the classical fractional harmonic maps first considered by Da Lio and Riviere. For $p_s \neq 2$ this is a new energy which has degenerate, non-local Euler-Lagrange equations. For the critical case, $p_s = n/s$, we show Holder continuity of these maps.

math.AP↗
arXiv2013

Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $π_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, $π_m(S^n)$. As an application we provide a new simplified proof of the fact that $π_n^{Lip}(H_n)\neq 0$, $n=1,2,...$, and we prove a new result that $π_{4n-1}^{Lip}(H_{2n})\neq 0$ for $n=1,2,...$ The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $W^{1,p}(M,H_{2n})$ when $dim M\geq 4n$ and $4n-1\leq p<4n$.

math.GT↗
arXiv2013

A note on zero sets of fractional sobolev functions with negative power of integrability

We extend a Poincaré-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincaré-style inequality.

math.AP↗
arXiv2013

A Game-Tree approach to discrete infinity Laplacian with running costs

We give a self-contained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tug-of-war games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce game-trees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi-)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for well-posedness.

math.AP↗