Tristan Riviere, Michael Struwe
Via Gauge theory, we give a new proof of partial regularity for harmonic maps in dimension m>2 into arbitrary targets. This proof avoids the use of adapted frames and permits to consider targets of "minimal" C^2 regularity. The proof we present moreover extends to a large class of elliptic systems of quadratic growth.
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We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $ε$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $ΣM$, and any compact Riemannian manifold $N$, we show an $ε$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $ψ\in W^{1,p}$ for some $p>2n/3$, th…
Regularity estimates on harmonic eigenmaps with arbitrary number of coordinates
本文重新审视曲面上调和映射的正则性估计对目标流形维数的独立性，研究映射到目标椭球体的 Laplace 调和特征映射，并收集工具处理几乎临界度量的收敛，为特征值无限组合的临界点的一般正则性理论提供初步步骤。
Quantitative stratification and optimal regularity for harmonic almost complex structures
本文通过定量分层方法改进了谐波近复结构的正则性理论，建立了最优正则性结果。
We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.
We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes r…
In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-manifold $N^n$ of an euclidian space ${\mathbb R}^m$ realizes a stationary varifold if and only if it is a smooth conformal harmonic map form $S$ into $N^n$.
In my previaou paper of K. Horihata, we have proposed a Ginzburg-Landau system with a time-dependent parameter and then passing to the limit we have constructed a harmonic heat flow into spheres. Thanks to this scheme, we establish a few energy inequalities of our flow: (i) monotonical inequalities and (ii) a reverse Poincare inequality at any boundary point. These inequalities (i) and (ii) derive the smaller estimates on the set on noncontinuouspoints for pur flow contrast to the former results. We refer to them by Y. Chen and Y. Chen, J. Li, F. H. Lin. Next we introduce two sufficient condit…
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harm…
We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from a Riemann surface to a sphere $§^n$. We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.
We show $C^{1,α}$-regularity for energy minimizing maps from a 2-dimensional Riemannian manifold into a Finsler space $(\R^n, F)$ with a Finsler structure $F(u,X)$.
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
Weighted Lojasiewicz inequalities and regularity of harmonic map flow
本文研究临界维数下调和映射流有限时间奇点，证明 Struwe 单调公式中量的 Lojasiewicz 不等式可推出体映射连续性与气泡树分解的无颈性质，并在目标为 $S^2$ 时证明了该不等式。
Optimal regularity of stable harmonic maps to spheres
本文研究到球面的稳定调和映射的正则性，确定其奇异集余维数的下界，证明结果最优，并给出n小于k时n维到k维球面非常数调和映射的非平凡指标界。
We discuss recent advances in the regularity problem of a variety of fluid equations and systems. The purpose is to illustrate the advantage of harmonic analysis techniques in obtaining sharper conditional regularity results when compared to classical energy methods.
In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider H^1_loc-maps u defined on a parabolic ball P\subset M\times R and with target manifold N, that have bounded Dirichlet-energy and Struwe-energy. We define a quantitative stratification, which groups together points in the domain into quantitative weakly singular strata S^j_{η,r}(u) according to the number of approximate symmetries of u at certain scales, and prove that their tubular neighborhoods have small volume, namely Vol(T_r(\cS^j_{η,r}(u))< Cr^{m+2-j-\eps}. In particular, th…
Optimal higher regularity for biharmonic maps via quantitative stratification
本文利用定量分层方法改进Breiner-Lamm关于双调和映射的几乎最优正则性结果，并得到极小化双调和映射的最优正则性。
We study Dirichlet problems for harmonic maps from a Riemannian $m$-manifold $(M,g)$ into a Finsler $n$-manifold $(N, F)$. We assume that the dimension of the source manifold $M$ is less than or equal to 4, and that the finsler structure $F(u,X)$ is given as F(u,X)= \sqrt{h_{ij}(u)X^i X^j + {\cal B}(u,X)}, (u\in N, X \in T_uN) where $(h_{ij})$ is a Riemannian metric and ${\cal B}(u,X)$ is a function on $TN$ with positive homogeneity of degree 2 with respect to $X$. Under these assumptions, an existence and interior regularity result will be given.
Regularity of minimizing $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$
本文证明从三维球到三维球面的极小p-调和映射在p>1.44时具有局部C^{1,α}正则性，填补了先前正则性区间的空白，并覆盖所有p≥2及次二次范围。
Boundary partial regularity for a class of Dirac-harmonic maps
本文研究一类满足特定能量单调不等式的耦合 Dirac-harmonic 映射，证明其在边界附近具有部分正则性。
Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - characterized by topological and analytic properties of tangent maps - is a topological manifold. In the special case of maps into the sphere ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-d…
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
We study the Sobolev regularity of $p$-harmonic functions. We show that $|Du|^{\frac{p-2+s}{2}}Du$ belongs to the Sobolev space $W^{1,2}_{loc}$, $s>-1-\frac{p-1}{n-1}$, for any $p$-harmonic function $u$. The proof is based on an elementary inequality.
Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality
该研究针对取值于3维球面的3维单位球上的极小$p$-调和映射，结合经典方法与新工具拓展了已知正则性的$p$值区间，还证明了二维区域下的精确Kato不等式。
A Note on Moving Frames along Sobolev Maps and the Regularity of Weakly Harmonic Maps
本文证明，对弱可微映射，梯度 Morrey 范数的小量条件保证拉回切丛被有限能量标架平凡化；对弱调和映射，BMO 半范数的小量条件保证完全正则性。
We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of ${\mathbb R}^{n+1}$ with bounded energy. This generalizes results obtained by Da Lio on the disc.
$\infty$-Harmonic maps are a generalization of $\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\infty$-harmonic maps from and into a sphere, quadratic $\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\infty$-harmonic maps between complex Euclidean spaces.
Sacks-Uhlenbeck type regularity for subcritical generalized $p$-harmonic maps into Homogeneous targets
本文定义到齐性目标的广义 $p$-调和映射，在次临界范围 $p>n$ 且能量一致有界时，证明 $p\searrow n$ 极限下的一致 $C^{1,\alpha}$ 正则性。
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 prove that if the minors of degree $k$ of a Sobolev map $\mathbb{R}^d \to \mathbb{R}^d$ are smooth then the map is smooth, when $k,d$ are not both even. We use this result to derive a simple, self-contained proof of the famous Liouville theorem for conformal maps, under the weakest possible regularity assumptions, in even dimensions which are not multiple of $4$. This is based on the approach taken by Iwaniec and Martin in [Acta Mathematica, 170(1):29--81, 1993]. We also prove the regularity of $W^{1,d/2}$ conformal maps between Riemannian manifolds, under the additional assumption of conti…
Regularity from $p$-harmonic potentials to $\infty$-harmonic potentials in convex rings
本文在任意凸环域中研究从p-调和势到∞-调和势的正则性，证明了∞-调和势的内部C^1正则性及部分Sobolev正则性。
In the seminal paper [Arch. Ration. Mech. Anal. 176 (2005), 351--361], Savin proved the $C^1$-regularity of planar $\infty$-harmonic functions $u$. Here we give a new understanding of it from a capacity viewpoint and drop several high technique arguments therein. Our argument is essentially based on a topological lemma of Savin, a flat estimate by Evans and Smart, % \cite{es11a}, $W^{1,2}_{loc}$-regularity of $|Du|$ and Crandall's flow for infinity harmonic functions.
We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the $p$-Laplace equation in the Heisenberg group $\mathbb{H}^1$ for $p>4 $.
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(Ω, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of stationary harmonic map and prove the partial regularity. We also discuss the global Lipschitz and piecewise $C^{1,α}$-regularity of harmonic maps from $(Ω, g)$ manifolds that support convex distance functions.
The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps …
We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,β}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $β\in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the singular set.
We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hoelder continuous derivatives. This gives an extension of a result by S. Hildebrandt et al concerning harmonic maps.
In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bounds currently available in literature for generic $p$. The main technique used in this paper is the quantitative stratification, which is based on the study of the approximate symmetries of the tangent maps of $f$. In this article, we generalize the study carried out in \cite{Ch…
We prove a sharp criterion on the decay of the tension of almost harmonic maps from degenerating surfaces that ensures that such maps subconverge to a limiting object that is made up entirely of harmonic maps.
Let $1<p\le4$ when $n=1$ and $1<p< 3+\frac{1}{n-1} $ when $n\ge2$. We obtain the second-order horizontal Sobolev $HW^{2,2}_{\rm loc} $-regularity of $p$-harmonic functions in the Heisenberg group $\mathbb H^n$. This improves the known range of $p$ obtained by Domokos and Manfredi in 2005.
We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.
We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.
We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are call…
In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.
We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].
Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) spaces
该研究证明了从n维Heisenberg群到CAT(0)空间的次椭圆调和映射具备局部Lipschitz连续性，拓展了已有Hölder正则性结果，为次椭圆调和映射的一般正则性理论提供了通用方法。
4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first differen…
Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary
本文证明了在域度量沿自由边界以$d^α$速率退化或奇异时，具有自由边界数据的弱平稳调和映照的部分正则性。
We give a direct harmonic approximation lemma for local minima of quasiconvex multiple integrals that entails their $\mathrm{C}^{1,α}$ or $\mathrm{C}^{\infty}$-partial regularity. Different from previous contributions, the method is fully direct and elementary, only hinging on the $\mathrm{L}^{p}$-theory for strongly elliptic linear systems and Sobolev's embedding theorem. Especially, no heavier tools such as Lipschitz truncations are required.
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.
Regularity of quasi-n-harmonic mappings into NPC spaces
本文证明了从欧氏区域到 Alexandrov 意义下非正曲率空间的拟 n-调和映射的局部 Hölder 连续性，并进一步推导出其在有界 Lipschitz 区域上的全局 Hölder 连续性。
Boundary regularity of optimal transport maps on convex domains
该研究针对凸域间二次代价最优运输映射的边界正则性，证明了非退化密度下势函数的$C^{1,1-\boldsymbol{\text{ε}}}$正则性、改进了边界$C^{1,α}$时的正则性、刻画了平面多面体逐点$C^{1,1}$正则性，并引入了刻画爆破齐次性的单调性公式。
Quantitative regularity for minimizing intrinsic fractional harmonic maps
本文研究极小化内蕴分数阶调和映射，结合修正Luckhaus引理与定量分层理论证明紧性、奇异集体积估计及改进的维数界。
A brief introduction to the regularity theory of minimising harmonic maps
本文综述极小调和映射正则性理论的核心结果，涵盖变分、正则性定理、爆破与奇异集分析，并延伸至平稳情形及近期进展。
We prove that local weak solutions of the orthotropic $p-$harmonic equation in $\mathbb{R}^2$ are $C^1$ functions.
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