arXiv subjects
Publications and source records attributed to Armin Schikorra.
Let $Ω\subset \mathbb{R}^n$ be an open set and $f_k \in W^{s,p}(Ω;\mathbb{R}^n)$ be a sequence of homeomorphisms weakly converging to $f \in W^{s,p}(Ω;\mathbb{R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is injective almost everywhere in the domain and the target. In this note we extend such results to the case $s\in(0,1)$ and $sp > n-1$. This in particular applies to $C^s$-Hölder maps.
We show that the operator \[ T_{K,s_1,s_2}f(z) := \int_{\mathbb{R}^n} A_{K,s_1,s_2}(z_1,z_2) f(z_2)\, dz_2 \] is a Calderon-Zygmund operator. Here for $K \in L^\infty(\mathbb{R}^n \times \mathbb{R}^n)$, and $s,s_1,s_2 \in (0,1)$ with $s_1+s_2 = 2s$ we have \[ A_{K,s_1,s_2}(z_1,z_2) = \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) \left (|x-z_1|^{s_1-n} -|y-z_1|^{s_1-n} \right )\, \left (|x-z_2|^{s_2-n} -|y-z_2|^{s_2-n}\right )}{|x-y|^{n+2s}}\, dx\, dy. \] This operator is motivated by the recent work by Mengesha-Schikorra-Yeepo where it appeared as analogue of the Riesz transforms for the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y) (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = f[φ]. \]
Let $Σ$ a closed $n$-dimensional manifold, $\mathcal{N} \subset \mathbb{R}^M$ be a closed manifold, and $u \in W^{s,\frac ns}(Σ,\mathcal{N})$ for $s\in(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if $π_n(\mathcal{N})=\{0\}$ then there exists a minimizing $W^{s,\frac ns}$-harmonic map homotopic to $u$. If $π_n(\mathcal{N})\neq \{0\}$, then we prove that there exists a $W^{s,\frac{n}{s}}$-harmonic map from $\mathbb{S}^n$ to $\mathcal{N}$ in a generating set of $π_{n}(\mathcal{N})$. Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when $\frac{n}{s} \neq 2$ one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing $W^{s,\frac{n}{s}}$-maps into manifolds.
We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy = \langle f, φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] For $s \in (0,1)$, $t \in [s,2s]$, $p \in [2,\infty)$, $K$ an elliptic, symmetric, Hölder continuous kernel, if $f \in \left (H^{t,p'}_{00}(Ω)\right )^\ast$, then the solution $u$ belongs to $H^{2s-t,p}_{loc}(Ω)$ as long as $2s-t < 1$. The increase in differentiability is independent of the Hölder coefficient of $K$. For example, our result shows that if $f\in L^{p}_{loc}$ then $u\in H^{2s-δ,p}_{loc}$ for any $δ\in (0, s]$ as long as $2s-δ< 1$. This is different than the classical analogue of divergence-form equations ${\rm div}(\bar{K} \nabla u) = f$ (i.e. $s=1$) where a $C^γ$-Hölder continuous coefficient $\bar{K}$ only allows for estimates of order $H^{1+γ}$. In fact, it is another appearance of the differential stability effect observed in many forms by many authors for this kind of nonlocal equations -- only that in our case we do not get a "small" differentiability improvement, but all the way up to $\min\{2s-t,1\}$. The proof argues by comparison with the (much simpler) equation \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz = \langle g,φ\rangle \quad φ\in C_c^\infty(\mathbb{R}^n). \] and showing that as long as $K$ is Hölder continuous and $s,t, 2s-t \in (0,1)$ then the "commutator" \[ \int_{\mathbb{R}^n} K(z,z) (-Δ)^{\frac{t}{2}} u(z) \, (-Δ)^{\frac{2s-t}{2}} φ(z)\, dz - c\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (φ(x)-φ(y))}{|x-y|^{n+2s}}\, dx\, dy \] behaves like a lower order operator.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.
We consider dispersion generalized nonlinear Schrödinger equations (NLS) of the form $i \partial_t u = P(D) u - |u|^{2 σ} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $σ\in \mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.
We prove an $ε$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named author's recent work on free boundary harmonic maps and choose a good frame directly at the free boundary. This leads to growth estimates, which, in the critical regime $p=n$, imply Hölder regularity of solutions. In the supercritical regime, $p < n$, we combine the growth estimate with the geometric reflection argument: the reflected equation is super-critical, but, under the assumption of growth estimates, solutions are regular. In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a consequence of a monotonicity formula we obtain partial regularity up to the boundary away from a set of $(n-p)$-dimensional Hausdorff measure.
Let $\mathcal{M}$ be a Riemannian $n$-manifold with a metric such that the manifold is Ahlfors-regular. We also assume either non-negative Ricci curvature, or that the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. We characterize BMO-functions $u: \mathcal{M} \to \mathbb{R}$ by a Carleson measure condition of their $σ$-harmonic extension $U: \mathcal{M} \times (0,\infty) \to \mathbb{R}$. We make crucial use of a $T(b)$ theorem proved by Hofmann, Mitrea, Mitrea, and Morris. As an application we show that the famous theorem of Coifman--Lions--Meyer--Semmes holds in this class of manifolds: Jacobians of $W^{1,n}$-maps from $\mathcal{M}$ to $\mathbb{R}^n$ can be estimated against BMO-functions, which now follows from the arguments for commutators recently proposed by Lenzmann and the second-named author using only harmonic extensions, integration by parts, and trace space characterizations.
We extend the well-known result that any $f \in W^{1,n}(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(Ω)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distributional sense. Along the way we also obtain extensions to fractional Sobolev spaces $W^{s,\frac{n}{s}}$ of the degree estimates known for $W^{1,n}$-maps with positive or non-negative Jacobian, such as the sense-preserving property.
We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the Möbius energy under Möbius transforms, which fails for all the other O'hara energies. We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.
We estimate the Hopf degree for smooth maps $f$ from $\mathbb{S}^{4n-1}$ to $\mathbb{S}^{2n}$ in the fractional Sobolev space. Namely we show that for $s \in [1 - \frac{1}{4n}, 1]$ \[ \left |{\rm deg}_H(f)\right | \lesssim [f]_{W^{s,\frac{4n-1}{s}}}^{\frac{4n}{s}}. \] Our argument is based on the Whitehead integral formula and commutator estimates for Jacobian-type expressions.
Recently, Lenzmann and the author observed how to obtain a large class of sharp commutator estimates by a combination of an integration by parts, an harmonic extension, and trace space estimates. In this survey we review this approach in three concrete examples: the Jacobian estimate by Coifman-Lions-Meyer-Semmes, the Coifman-Rochberg-Weiss commutator estimate for Riesz transforms, and a Kato-Ponce-Vega-type inequality.
We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(Ω,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely \[\mathcal{H}^{n-3}(\textrm{sing} u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d\mathcal{H}^{n-1};\] (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.
We consider limits of weakly converging $W^{1,2}$-maps $Φ_k$ from a ball $B \subset \mathbb{R}^2$ into $\mathbb{R}^3$ which are conformal immersions. Under the assumption that a normal curvature term is small, namely if for the normal map $u$ we have for some $s \in (\frac{1}{2},1)$ $$ \int_{B} \int_{B} \left | \frac{u_k(x) \wedge u_k(y)}{|x-y|^{s}} \right|^{\frac{2}{s}}\, \frac{dx\, dy}{|x-y|^{2}} < \varepsilon $$ then we show that we can either pass to the limit and obtain an almost everywhere immersion $Φ$ or $Φ$ collapses and is constant. This is in the spirit of the results by T. Toro, and S. Müller and V. Sverak, and F. Hélein, who obtained similar statements under the stronger assumptions that the second fundamental form is bounded (but also stronger result: a locally bi-Lipschitz parametrization). The fractional normal curvature assumption is vaguely reminiscent of curvature energies such as the scaling-invariant limits of tangent-point energies for surfaces as considered by Strzelecki, von der Mosel et al. and we hope that eventually the analysis in this work can be used to define weak immersions with these kind of energy bounds.
We prove interior $C^{1,α}$-regularity for solutions \[ - Λ\leq F(D^2 u) \leq Λ\] where $Λ$ is a constant and $F$ is fully nonlinear, 1-homogeneous, uniformly elliptic. The proof is based on a reduction to the homogeneous equation $F(D^2u) = 0$ by a blow-up argument -- i.e. just like what is done in the case of viscosity solutions $F(D^2 u) = f$ for $f \in L^\infty$. However it was not clear to us that the above inequality implies $F(D^2 u) = f$ for some bounded $f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on $C^{1,α}$-regularity for viscosity inequalities. So we thought this result might be worth recording.
We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash \mathbb{B}^N(0)$}. \] Our main result is $C^{1,α}$-regularity for any $α< 1$. Technically, we split the map $u=λv$, where $v: \mathbb{B}^2 \to \mathbb{S}^{N-1}$ is the vectorial component and $λ= |u|$ the scalar component measuring the distance to the origin. While $v$ satisfies a weighted harmonic map equation with weight $λ^2$, $λ$ solves the obstacle problem for \[ \int_{\mathbb{B}^2} |\nabla λ|^2+λ^2 |\nabla v|^2, \quad \mbox{subject to $λ\geq 1$}. \] where $|\nabla v|^2 \in L^1(\mathbb{B}^2)$. We then play ping-pong between the increases in the regularity of $λ$ and $v$ to obtain finally the $C^{1,α}$-result.
The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions. We report on some recent progress in the Analysis of the Hölder topology of the Heisenberg group, some related and some unrelated to density questions for Sobolev maps into the Heisenberg group. In particular we describe the main ideas behind a result by Hajłasz, Mirra, and the author regarding Gromov's conjecture, which is based on the linking number. We do not prove or disprove the Gromov Conjecture.
We prove higher Hölder regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0 N/(s\,p)$, which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.