Paola Loreti, Daniela Sforza
Our goal is to establish a hidden regularity result for solutions of time fractional Petrovsky system where the Caputo fractional derivative is of order $α\in(1,2)$. We achieve such result for a suitable class of weak solutions.
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The purpose of this paper is to establish the regularity the weak solutions for a nonlinear biharmonic equation.
We study the regularity of solutions to the obstacle problem for the parabolic biharmonic equation. We analyze the problem via an implicit time discretization, and we prove some regularity properties of the solution.
We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.
We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.
We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $Σ$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $Σ$ is 1-rectifiable.
We consider critical points $u:Ω\to N$ of the bi-energy \[ \int_Ω|Δu|^2\,d x, \] where $Ω\subset\mathbb{R}^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\mathbb{R}^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(Ω,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the pr…
In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible (NTA) domain, we derive the $C^{1,α}$-regularity of the free boundary in a small ball centered at the origin. From the $C^{1,α}$-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $ C^{3,α}$ up to the free boundary, and therefore $C^{2,1} $. In the end we study an exa…
For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $Ω\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $λ\in [0,1]$ we introduce a $λ$-relaxed energy $\H_λ$ for the Hessian energy for maps in $W^{2,2}(Ω,S^4)$ so that each minimizer $u_λ$ of $\H_λ$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_λ$ for $λ\in [0,1)$.
We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $Ω\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth boundary data, we show that if, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $Σ\subset\barΩ$, with $H^{n-4}(Σ)=0$, such that $u\in C^\infty(\barΩ\setminusΣ, N)$.
This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic…
Regularity and uniqueness of the heat flow of biharmonic maps
本文在重整化总能量的小性条件下，建立了到单位球面及任意无边界紧致黎曼流形的双调和映射热流的正则性与唯一性，并证明了Hessian能量凸性及时间无穷大处的唯一极限。
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 …
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 consider a two obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions.
In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $Δ^2.$ With this characterizations, we prove the boundedness of the Bessel potentials, fractional integrals, fractional powers, Riesz transforms and multipliers of Laplace transforms type associated to $Δ^2$ on Lipschitz spaces and homogeneous Lipschitz spaces. The proofs of these results need the language of semigroups in an essential way.
A biharmonic analogue of the Alt-Caffarelli problem
该研究针对Dirichlet和Navier边界条件下的Alt-Caffarelli问题双调和类似问题，证明了极小元的存在性、基本性质与$C^{1,α}$正则性，得到Navier问题径向边界数据下的对称性结果，并验证了该正则性的最优性。
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 study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.
We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ Δ^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \int_{\mathbb{R}^N}|u|^2\log |u|\,dx\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|Δu|^2\,dx \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\ma…
Long-time reduction for the biharmonic nonlinear Schrödinger equation
本文证明在一维高阶色散消失极限下，双调和非线性薛定谔方程在中间正则性区间可长时约化为三次非线性薛定谔方程，并给出指数型逼近界，方法基于正则性保持。
Global results for a Cauchy problem related to biharmonic wave maps
本文在尺度临界 Besov 空间中证明了一类带非泛型二次非线性项的导数双调和波方程对小初值的全局存在性，并由此得到双调和波映射方程在包括球面在内的一类目标流形上的小数据全局正则性。
We study a fractional diffusion problem in the divergence form in one space dimension. We define a notion of the viscosity solution. We prove existence of viscosity solutions to the fractional diffusion problem with the Dirichlet boundary values by Perron's method. Their uniqueness follows from a proper maximum principle. We also show a stability result and basic regularity of solutions.
A Hamilton-Jacobi equation with Caputo's time-fractional derivative of order less than one is considered. The notion of a viscosity solution is introduced to prove unique existence of a solution to the initial value problem under periodic boundary conditions. For this purpose comparison principle as well as Perron's method is established. Stability with respect to the order of derivative as well as the standard one is studied. Regularity of a solution is also discussed. Our results in particular apply to a linear transport equation with time-fractional derivatives with variable coefficients.
The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for continuously differentiable functions. Accordingly, in the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of the smooth functions that are too narrow. In this paper, we introduce a suitable definition of the Caputo derivative in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fra…
We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order $α\in (0,1)$ cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case $α=1$, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive C…
This paper is concerned with an inverse source problem for the stochastic biharmonic operator wave equation. The driven source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The well-posedness of the direct problem is examined in the distribution sense and the regularity of the solution is discussed for the given rough source. For the inverse problem, the strength of the random source, involved in the principal symbol of its covariance operator, is shown to be uniquely determined by a single realizati…
This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Hölder continuous.
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
Time-dependent fractional-derivative problems $D_t^δu + Au = f$ are considered, where $D_t^δ$ is a Caputo fractional derivative of order $δ\in (0,1)\cup (1,2)$ and~$A$ is a classical elliptic operator, and appropriate boundary and initial conditions are applied. The regularity of solutions to this class of problems is discussed, and it is shown that assuming more regularity than is generally true---as many researchers do---places a surprisingly severe restriction on the problem.
In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $λ\in \mathbb R$, $d\in \mathbb N$, $0<s<\min \{2+\frac{d}{2},\frac{3}{2}d\}$ and $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $H^{s}(\mathbb R^{d})$ if $\frac{8-2b}{d}<σ< σ_{c}(s)$ and the initial data is sufficiently small, where $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and…
In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.
Motived by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm-Riviere system $$Δ^{2}u=Δ(V\cdot\nabla u)+{\rm div}(w\nabla u)+(\nablaω+F)\cdot\nabla u+f$$ in dimension four, with an inhomogeneous term $f$ which belongs to some natural function space. We obtain optimal higher order regularity and sharp Holder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mapp…
$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions
本文建立了所有超临界维数下带反对称势的四阶椭圆系统的最优$L^p$-正则性理论，改进了Struwe的Hölder正则性定理，并扩展了调和映射的$L^p$-正则性理论。
Lipschitz Stability for an Inverse Problem of Biharmonic Wave Equations with Damping
本文研究双曲波方程中变密度系数和初始位移的联合反演问题，通过乘子技术推导观测不等式，得到逆问题的Lipschitz稳定性估计，证明双曲结构增强了参数识别的稳定性。
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearit…
In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation \[iu_{t} +Δ^{2} u=λ|x|^{-b}|u|^σu,~u(0)=u_{0} \in H^{s} (\mathbb R^{d}),\] where $d\in \mathbb N$, $s\ge 0$, $0<b<4$, $σ>0$ and $λ\in \mathbb R$. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is locally well-posed in $H^{s}(\mathbb R^{d})$ if $d\in \mathbb N$, $0\le s <\min \{2+\frac{d}{2},\frac{3}{2}d\}$, $0<b<\min\{4,d,\frac{3}{2}d-s,\frac{d}{2}+2-s\}$ and $0<σ< σ_{c}(s)$. Here $σ_{c}(s)=\frac{8-2b}{d-2s}$ if $s<\frac{d}{2}$, and $σ_{c…
Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives
本文针对具有Caputo导数的多项时间分数扩散方程的反源问题，通过过定条件重建未知源项，并利用Mittag-Leffler函数的谱表示和渐近展开，推导了经典解的存在性和唯一性条件。
In the present work, we discuss a unique solvability of an inverse-source problem with integral transmitting condition for time-fractional mixed type equation in a rectangular domain, where the unknown source term depends on space variable only. The method of solution based on a series expansion using bi-orthogonal basis of space corresponding to a nonself-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. Influence of transmitting condition on the solvability of the problem is show…
In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $ Ξ(z)=e^{|z|^p}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution …
Trace operators for Riemann--Liouville fractional equations
本文概述常用分数阶导数后，研究含Riemann-Liouville导数的分数阶时间波动方程的适定性、正则性等问题及与Caputo导数的对偶关系，通过Mittag-Leffler函数展开解进行分析。
The main purpose of this paper is to show the global stabilization and exact controllability properties for a fourth order nonlinear fourth order nonlinear Schrödinger system: $$i\partial_tu +\partial_x^2u-\partial_x^4u=λ|u|^2u,$$ on a periodic domain $\mathbb{T}$ with internal control supported on an arbitrary sub-domain of $\mathbb{T}$. More precisely, by certain properties of propagation of compactness and regularity in Bourgain spaces, for the solutions of the associated linear system, we show that the system is globally exponentially stabilizable. This property together with the local exa…
We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first…
We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-Δu = Q \; \mathcal{H}^{n-1} \llcorner Γ$ in a domain $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $Γ\subset \subset Ω$ is a compact (Lipschitz) manifold and $Q \in L^\infty(Γ)$. We also discuss optimality and necessity of the assumptions on $Q$ and $Γ$. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.
Large time behavior for the hyperbolic-parabolic coupled system with the regularity-loss structure
本文研究全空间中源自广义热弹性耦合系统的双曲-抛物耦合系统的大时间行为，通过对角化与WKB分析得到新的正则性损失型大时间渐近轮廓及最优衰减估计，并发现带Riesz位势耗散的波动方程是其大时间近似模型。
In this paper, we consider a nonlinear beam equation with the p-biharmonic operator, where $1 < p < \infty$. Using a change of variable, we transform the problem into a system of differential equations and prove the existence, uniqueness and regularity of the weak solution by applying the Lax-Milgram theorem and classical results of functional analysis. We investigate the discrete formulation for that system and, with the aid of the Brouwer theorem, we show that the problem has a discrete solution. The uniqueness and stability of the discrete solution are obtained through classical methods. Af…
Regular Strichartz estimates in Lorentz-type spaces with application to the $H^s$-critical inhomogeneous biharmonic NLS equation
本文利用Lorentz型空间中的正则Strichartz估计，在更弱正则性条件下建立$H^s$-临界非齐次双调和NLS方程的局部适定、小数据全局适定与散射结果。
In a recent article \textit{"Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane", to appear on Pacific Journal of Mathematics [15]}, the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [15] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure $\mathcal{G}$. Precisely, consider $\mathcal{G}$ composed by $N$ e…
Sharp Morrey regularity theory for a fourth order geometrical equation
该研究延续前人工作，推导了最低正则性假设下四阶非齐次Lamm-Rivière方程弱解的精确Morrey正则性理论，深化了Lᵖ型正则性理论并将二阶问题结果推广至四阶情形。
This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schrödinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+εΔu+δA u+λ|u|^αu=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $Δ^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropic dispersion), and $α, ε, λ$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also …
Non-local time problem for the Rayleigh--Stokes type fractional equations
本文研究了Rayleigh-Stokes方程中Caputo导数的非局部时间问题，通过拉普拉斯变换和傅里叶方法分析了初始边界值问题的适定性。
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $γ\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimat…
Inverse problem for multi-term fractional parabolic equation in two dimensional space, involving m + 1 Caputo fractional derivatives in time, is investigated. Presence of nonlocal boundary conditions leads to a non-self-adjoint spectral problem. A bi-orthogonal system of functions is used to construct the solution that involves double infinite series. Properties of multinomial Mittag-Leffler function and eigenfunctions are used to prove the classical nature of the solution under certain regularity conditions on the given datum.
Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity
本文研究混合阶拟线性椭圆方程，建立紧加权嵌入与变分框架，证明解的存在性与多重性，并利用势估计和自举方法获得局部正则性及Pohozaev型恒等式。
Resolvent Approach to Atangana--Baleanu Evolution Equations: Laplace Symbols, Mild Solutions, and Regularity
本文提出基于Atangana-Baleanu导数的解析方法，通过引入分数解析和Mittag-Leffler核，建立了AB型演化方程的弱解表示，并推导了稳定性与正则性估计。
Superdiffusive fractional dynamics: Unveiling regularity results in systems with general positive self-adjoint operators
该研究针对含一般正自伴算子的时间分数阶柯西问题，在不假设算子紧预解式的条件下，证明了弱与强能量解的存在性和正则性，拓展了此前相关研究结果，可应用于记忆效应等超扩散现象建模。
We present a maximal $L_{q}(L_{p})$-regularity theory with Muckenhoupt weights for the equation \begin{equation}\label{eqn 01.26.16:00} \partial^α_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in\mathbb{R}^{d}. \end{equation} Here, $\partial^α_{t}$ is the Caputo fractional derivative of order $α\in(0,2)$ and $a^{ij}$ are functions of $(t,x)$. Precisely, we show that \begin{equation*} \begin{aligned} &\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-Δ)^{γ/2}u_{xx}(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt \\ &\quad \leq N \int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-Δ)^{γ/2}f(t,x)|^{p}w_…
Polyharmonic equations involving surface measures
本文将二阶方程带曲面测度项的正则性结果推广到多调和方程，证明其解的最优 $W^{2m-1,\infty}$ 正则性，并应用于二维双调和 Alt-Caffarelli 问题，得到最优 $W^{3,\infty}$ 正则性。
Regularity of solutions to space--time fractional wave equations: a PDE approach
本文研究了涉及分数次幂的演化方程，探讨了其解的正则性，结合PDE方法给出了存在性、唯一性和能量估计，并推导了时间与空间的正则性估计，指出传统数值分析文献中的假设存在问题。
Inverse problems for time-fractional Schrödinger equations
该研究针对含Caputo导数的时间分数阶薛定谔方程反问题，通过弱化初始数据正则性，从正勒贝格测度集出发，证明了改进的唯一性结果。
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