Jinsil Lee, Yong-Hoon Lee
In this work, we study the regularity of positive solutions for nonlinear fractional differential equation with a singular weight. We define the new Banach space and use this space to show the regularity. We also give an example with a singular weight which may not be in $L^1.$
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In this report we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in $\mathbb{R}^{1}$. The analysis is performed in the weighted Sobolev spaces, $H_{(a , b)}^{s}(\mathrm{I})$. Three different characterizations of $H_{(a , b)}^{s}(\mathrm{I})$ are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters $α$ and $r$ defining the fractional diffusion operator. Additiona…
We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function maps $L^{p}$ into a first order Sobolev space in dimensions $n \geq 5$.
We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.
The generalized (or coupled) Abel equations on the bounded interval have been well investigated in H$\ddot{\text{o}}$lderian spaces that admit integrable singularities at the endpoints and relatively inadequate in other functional spaces. In recent years, such operators have appeared in BVPs of fractional-order differential equations such as fractional diffusion equations that are usually studied in the frame of fractional Sobolev spaces for weak solution and numerical approximation; and their analysis plays the key role during the process of converting weak solutions to the true solutions. Th…
Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms
针对含时间分数阶导数与非自治形式的 J.L. Lions 问题，本文在 Dini 型连续条件下证明极大 $L_p$-正则性，并以反例揭示其所需的最低 Hölder 正则性。
In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their Hölder exponent everywhere (using several definitions) and compute the associated spectrum of singularities. We also show that these functions satisfy an approximate functional equation, and use it to discuss the graphs of "Riemann's example" and of fractional integrals of cusp forms for $Γ_0(N)$. We include some computer plots.
Stochastic maximal $L^p$-regularity for non-autonomous evolution equations with fractional derivative in UMD spaces
本文研究UMD空间中带广义分数阶导数的非自治随机演化方程的极大$L^p$-正则性，通过时间加权空间和正则-奇异分解获得适定性、时空正则性及随机极大$L^p$-正则性，并应用于非自治随机扩散方程和随机分数阶反应-扩散SIR模型。
In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus $\mathbb{T}^n$ from the fractional Laplacian on $\mathbb{R}^n$. Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that $L^2$ functions on the torus can not be identified with $L^2$ functions on $\mathbb{R}^n$. The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise…
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…
Let $L=-Δ+V$ be a Schrödinger operator, where the potential $V$ belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup $\{e^{-t{L}^α}\}_{t>0}, α>0$, associated with ${L}$. By the aid of the fundamental solution of the heat equation: $$\partial_{t}u+L u=\partial_{t}u -Δu+Vu=0,$$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K^{L}_{α,t}(\cdot, \cdot)$, respectively. This method is independent of the Fourier transform, and can be applied to the second order differential operators whose heat kernel…
Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $Ω$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~Ω,\\ u=0,&\hbox{on}~\partialΩ, \end{cases}\quad \hbox{and}\quad \begin{cases} L^su=f,&\hbox{in}~Ω,\\ \partial_Au=0,&\hbox{on}~\partialΩ. \end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and $\partial_Au$ is the conormal derivative of $u$ with respect to the coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prov…
We study the obstacle problem for a nonlocal, degenerate elliptic Monge--Ampère equation. We show existence and regularity of a unique classical solution to the problem and regularity of the free boundary.
A tourist's guide to regularity structures and singular stochastic PDEs
该研究给出了正则结构的基本分析与代数特征及其在奇异随机偏微分方程研究中应用的自洽性阐述，为相关领域提供入门指引。
This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection reaction equation (FDARE) with variable coefficients on the bounded interval. The existence of the true (classical) solution together with norm estimation is established and the precise regularity bound is found; also, the structure of the solution is unraveled, capturing the essence of regularity, singularity, and other features of the solution. The key analy…
We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: $\Box u = a |\partial_t u|^2+b|\nabla_x u|^2$, $u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}$, $\partial_t u(0,x)=u_1(x)\in H^{s-1}_{\mathrm{rad}}$. It has been known that the problem is well-posed for $s\ge 2$ and ill-posed for $s<3/2$. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for $s>3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent …
For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein--Uhlenbeck equation $$\begin{cases} (-Δ+x\cdot\nabla)^su=f&\hbox{in}~Ω\\ u=0&\hbox{on}~\partialΩ, \end{cases}$$ where $Ω$ is a possibly unbounded open subset of $\mathbb{R}^n$, $n\geq2$. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel $L^p$ and $L^p(\log L)^α$ regularity estimates in terms of the datum $f$ are obtain…
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