Huyuan Chen, Ying Wang
Our concern in this paper is to study the qualitative properties for harmonic functions related to the fractional Laplacian. Firstly we classify the polynomials in the whole space and in the half space for the fractional Laplacian defined in a principle value sense at infinity. Secondly, we study the fractional harmonic functions in half space with singularities on the boundary and the related distributional identities.
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We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.
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.
This article addresses the regularity issue for minimizing fractional harmonic maps of order $s\in(0,1/2)$ from an interval into a smooth manifold. Hölder continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then Hölder continuity holds everywhere.
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.
We prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on the whole real line. In both cases we use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp estimates obtained via commutator techniques.
On a Fractional Schrödinger equation in the presence of Harmonic potential
本文研究了调和势存在下分数阶薛定谔方程基态解的存在性与驻波轨道稳定性，通过数值模拟比较揭示了分数阶幂、非线性幂及调和势各项对系统动力学的影响。
We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.
In this paper, we continue to study the fractional harmonic gradient flow on $S^{n-1}$ taking values in a general closed manifold $N \subset \mathbb{R}^n$, addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivière in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a ge…
This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. …
We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $Ω\subset \mathbb R^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-Δ)^s u = 0 \quad \mbox{in} \ Ω, $$ where $(-Δ)^s$ denotes the fractional Laplacian of order $2s \in (0,2)$. We use the viscosity solution's theory and Perron's method to construct harmonic functions with zero exterior condition in $\bar Ω^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in Ω}\mathrm{dist}(x, \parti…
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 global regularity theory for shpere-valued fractional harmonic maps
本文针对球值分数阶调和映射，利用 Cheeger-Naber 定量微分方法建立奇异集的定量分层理论，从而获得全局正则性估计。
$C^{1+α}$ regularity for fractional $p$-harmonic functions
该研究证明了当 $p$ 位于 $[2,2/(1-s))$ 时，分数阶 $p$-Laplace 方程的解具有内部 $C^{1,α}$ 正则性。
We prove continuity on domains up to the boundary for n/2-polyharmonic maps into manifolds. Technically, we show how to adapt Helein's direct approach to the fractional setting. This extends a remark by the author that this is possible in the setting of Riviere's famous regularity result for critical points of conformally invariant variational functionals. Moreover, pointwise behavior for the involved three-commutators is established. Continuity up to the boundary is then obtained via an adaption of Hildebrandt and Kaul's technique to the non-local setting.
Hölder extension for fractional Laplacian
本文刻画了分数阶调和延拓全局Hölder连续的尖锐边界条件，证明依赖于分数阶调和测度衰减估计与补集的均匀分数阶丰满性。
A positive homogeneous fractional harmonic function in a Lipschitz cone
本文在Lipschitz锥中构造了求解分数拉普拉斯方程$(-Δ)^{s}u=0$的正齐次函数，证明了其在常数乘法因子下的唯一性及其度数$0<α<2s$的齐次性。
In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlineari…
We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $Ω$ with two different boundary conditions at the same time: the shape of the solution outside $Ω$ and a weighted limit to the boundary. We allow the nonlinearity to be positive or negative and we look for solutions blowing up at the boundary. Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative nonlin…
Fine regularity of fractional harmonic maps and applications
本文研究分数调和映射的正则性问题，通过非局部方程理论改进了已知结果，并应用于具有自由边界的调和映射及热流的可微性研究。
We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^β}$, $t, β>0$, associated with the harmonic oscillator $H=-Δ+ |x|^2$. We then prove some local and global wellposedness results for nonlinear fractional heat equations.
In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$ must be constant.} A new idea is employed to prove this result, which is completely different from the previous ones in deriving Liouville theorems. For any given hyper-plane in $\mathbb{R}^n$, we show that $u$ is symmetric about the plane. To this end, we established a {\em maximum principle} for anti-symmetric functions on any half space. We believe that th…
The uniqueness of the positive ground state solutions of fractional Shrodinger equations with a harmonic potential has not been covered by the breakthrough method developed in [1, 2]. It has remained an open question for years. [3] and [5] were quite recently able to prove the existence of the ground state solutions and to derive important properties but they failed to address the uniqueness. [3] left it as an open question, and [5] run numerical simulations that were in favor of the uniqueness. Very recently, the authors of this note developed a general and unified method to prove the uniquen…
In this paper we consider critical points of the following nonlocal energy {equation} {\cal{L}}_n(u)=\int_{\R^n}| ({-Δ})^{n/4} u(x)|^2 dx\,, {equation} where $u\colon H^{n/2}(\R^n)\to{\cal{N}}\,$ ${\cal{N}}\subset\R^m$ is a compact $k$ dimensional smooth manifold without boundary and $n>1$ is an odd integer. Such critical points are called $n/2$-harmonic maps into ${\cal{N}}$. We prove that $Δ^{n/2} u\in L^p_{loc}(\R^n)$ for every $p\ge 1$ and thus $u\in C^{0,α}_{loc}(\R^n)\,.$ The local Hölder continuity of $n/2$-harmonic maps is based on regularity results obtained in \cite{DL1} for nonlocal…
Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential
本文针对带调和势的分数阶非线性椭圆方程，证明了满足特定参数条件下正解的唯一性，解决了相关文献中提出的公开问题。
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.
Local Regularity of very weak $s$-harmonic functions via fractional difference quotients
本文提出通过改进局部可和性、适配分数阶差商方法消除积分核奇异性并结合标准椭圆正则性方法，新证单位球中极弱s-调和函数的实解析性，相关差商技术为首次提出。
As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of balls, namely whether a set for which all harmonic functions satisfy the mean value property is necessarily a ball. This question was investigated by several authors, and was finally elegantly, completely and positively settled by Ülkü Kuran, with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, N…
Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential
本文研究了分数非线性椭圆方程在调和势下的地面态唯一性问题，通过证明唯一性解决了相关开放问题。
We investigate the existence, uniqueness, and $L^1$-contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface. To settle the existence, we reformulate the equation as a local problem on a semi-infinite cylinder, regularise the porous medium nonlinearity and truncate the cylinder. Then we pass to the limit first in the truncation parameter and then in the nonlinearity, and the identification of limits is done using the theory of subdifferentials of convex functionals. In order to facilitate all of this, we begin by studying (in the setting…
Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the \emph{…
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 fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy
本文提出了半Dirichlet能量平稳性的弱形式，用分数阶Hopf微分表达，并证明共形调和映射与平稳点的调和延拓等价，同时推导了Noether定理和Pohozaev恒等式。
In this paper we discuss the relation between non-homogeneous nonlinear fractional diffusive equations and the Schrodinger equation with time-dependent harmonic potential. It is well known that the Cole-Hopf transformation allows to linearize non-homogeneous nonlinear diffusive equations (NHNDEs) into a Schrodinger-type equation with time-dependent potential. We first discuss the utility of the results about time-dependent harmonic oscillator to build explicit solution of such non-homogeneous nonlinear partial differential equations. In particular, we recall that starting from a trial polynomi…
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 f…
The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences.
We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wente's inequality by L. Tartar, instead of Besov-space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.
The existence and nonexistence of $λ$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $Ω$ is zero, then there is no bounded $λ$-harmonic function of $Ω$ for $λ\in [0,λ_1(\mathbb{H}^n)]$, where $λ_1(\mathbb{H}^n)=(n-1)^2/4$. For these domains, we have comparison principle and some maximum principle. Conversely, for any $s>(n-1)/2,$ we prove the existence of domains with asymptotic boundary of dimension $s$ for which there are bounded $λ_1$-harmonic functions that decay exponential…
We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every $α$-harmonic function bounded either above or below in all of $R^n$ must be constant.
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.
Elementary Brezis-Browder type results and Representation formulae for s-harmonic functions
本文证明了分数Sobolev空间中的Brezis-Browder型结果及s-调和函数的定量估计，并给出了分数泊松方程分布解为特定形式的充分条件。
A Lions' type formula for some reproducing kernel Hilbert spaces of fractional harmonic functions
本文研究了分数阶调和函数的再生核希尔伯特空间，推导出分数泊松公式并应用其获得狮子型公式，该公式与Hadamard变分公式相似，尽管算子阶非为2。
This master thesis is based on the paper "Sharp commutator estimates via harmonic extensions" by Lenzmann and Schikorra, in which they proposed a method to prove estimates for commutators involving Riesz transforms, fractional Laplacians and Riesz potentials, see arXiv:1609.08547. These proofs only involve harmonic extensions to the upper half-space and integration by parts next to some elementary transfromations, since the deeper theory is concentrated in a variety of trace characterization results which can be used as a blackbox. In the first half of this thesis, after collecting some elemen…
In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator $H^σ=(-Δ+|x|^2)^σ$, $0<σ<1$. More precisely, a new class of smooth functions $C^{k,α}_H$ is defined, in which we study the action of $H^σ$. It turns out that these spaces are the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite-Riesz transforms with the Hölder spaces $C^{k,α}_H$ is needed, that we believe of independent interest. The parallel results for the fractional powers of the Laplacian $(-Δ)^σ$ were applied by Caf…
We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms.
We give an appropriate version of the result in the paper by Dipierro, Savin, and Valdinoci for different, not necessarily fractional, powers of the Laplacian.
The dimension of harmonic measure on some AD-regular flat sets of fractional dimension
本文研究分数维AD正则平坦集的调和测度维数问题，证明当$s$-AD正则紧集含于超平面或n维C¹流形且s∈[n-1/2,n)时，其补域调和测度的Hausdorff维数严格小于集合自身的Hausdorff维数s。
Let $L=Δ^{α/2}+ b\cdot\nabla$ with $α\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.
We study gradient estimates of $q$-harmonic functions $u$ of the fractional Schr{ö}dinger operator $Δ^{α/2} + q$, $α\in (0,1]$ in bounded domains $D \subset \R^d$. For nonnegative $u$ we show that if $q$ is H{ö}lder continuous of order $η> 1 - α$ then $\nabla u(x)$ exists for any $x \in D$ and $|\nabla u(x)| \le c u(x)/ (\dist(x,\partial D) \wedge 1)$. The exponent $1 - α$ is critical i.e. when $q$ is only $1 - α$ H{ö}lder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for $α\in (1,2]$ under the assumption that $q$ belongs to the Kato class $\calJ^{α- 1}$. …
We consider Backus's problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth's surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field's intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is…
Gradient regularity for $(s,p)$-harmonic functions
本文研究$(s,p)$-调和函数的局部正则性，证明其弱可微、弱梯度任意阶局部可积，并得到任意Hölder连续性及分数阶可微性，且估计在$s\to1$时稳定并恢复$p$-调和函数的已知正则性。
We introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.
One-dimensional half-harmonic maps into the circle and their degree
本文研究一维半调和映射到圆，证明在不同同伦类中存在极小化分数Dirichlet能量的第二个映射，基于度数分析和Brezis-Coron估计，并指出与二维情形的差异。
In this paper, we prove the following result. Let $α$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-Δ)^{α/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\ \displaystyle\underset{|x| \to \infty}{\underline{\lim}} \frac{u(x)}{|x|^γ} \geq 0 , \end{array} \right. $$ for some $0 \leq γ\leq 1$ and $γ< α$. Then $u$ must be constant throughout $\mathbb{R}^n$. This is a Liouville Theorem for $α$-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using p…
Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains
该研究针对Reifenberg平坦区域，证明了分数阶p-调和函数商的边界哈纳克不等式与赫尔德连续性，通过系列估计和变形方法将相关结果从能量类扩展到尾类。
On the Harnack inequality for antisymmetric $s$-harmonic functions
本文在一般区域中证明了反对称 $s$-调和函数及带零阶项分数阶方程解的 Harnack 不等式，该结果可与移动平面法结合以获得由分数阶 Laplacian 驱动的半线性方程对称性和超定问题的定量稳定性。
The operator square root of the Laplacian $(-\lap)^{1/2}$ can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.
Fractional De Giorgi conjecture in dimension 2 via complex-plane methods
本文利用复平面方法，结合 Farina 方法和 Caffarelli-Silvestre 的 $s$-调和延拓，为二维全指数范围 Allen-Cahn 方程的分数阶 De Giorgi 猜想提供了新证明，并建立了带 Neumann 边界条件的加权椭圆方程有限能量弱解的表示公式。
We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations. Furthermore, an example of an $s$-harmonic function, the harmonic extension and some insight on a fractional version of a classical conjecture formulated by De Giorgi are presented. Although this book aims at gathering some introductory material on the applications of the fractional…
The aim of this paper is to get the boundedness of certain multi-sublinear operators generated by multilinear fractional integral operators on the product generalized local Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutators of multilinear operators generated by local campanato functions and multilinear fractional integral operators are also bounded on the product generalized local Morrey spaces.
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.
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