We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Holder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work [12] by leaving the framework of distributed order fractional time derivatives and considering a general PC kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.
We prove the local boundedness and the local Hölder continuity of weak solutions to nonlocal equations with variable orders and exponents under sharp assumptions.
The solution of time fractional partial differential equations in general exhibit a weak singularity near the initial time. In this article we propose a method for solving time fractional diffusion equation with nonlocal diffusion term. The proposed method comprises L1 scheme on graded mesh, finite element method and Newton's method. We discuss the well-posedness of the weak formulation at discrete level and derive \emph{a priori} error estimates for fully-discrete formulation in $L^2(Ω)$ and $H^1(Ω)$ norms. Finally, some numerical experiments are conducted to validate the theoretical findings…
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) $β$-Hölder continuous for some $β\in (1-s,\,1]$, where $s \in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $β$-Hölder continuous.
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 give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
We study local boundedness and Hölder continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right hand side.
We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.
In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonloca…
We prove local boundedness and Hölder continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} + a(x,y)\frac{|v(x)-v(y)|^q}{|x-y|^{n+tq}}\, dxdy, \] where $0<s\le t<1<p \leq q<\infty$ and $a(\cdot,\cdot) \geq 0$. For such regularity results, we identify sharp assumptions on the modulating coefficient $a(\cdot,\cdot)$ and the powers $s,t,p,q$ which are analogous to those for local double phase problems.
We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as …
Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation
Taras Shevchenko National University（塔拉斯·舍甫琴科国立大学）；Johann Radon Institute（约翰·拉德恩研究所）；Istituto Italiano di Tecnologia（意大利理工学院）；Institute of Applied Mathematics and Mechanics of NAS of Ukraine（乌克兰国家科学院应用数学与力学研究所）；S.P. Timoshenko Institute of Mechanics of NAS of Ukraine（乌克兰国家科学院S.P. 什捷潘诺夫力学研究所）
Some Unexplored Topics in The Reconstruction of Scalar Parameters of Subdiffusion
Istituto Italiano di Tecnologia（意大利理工学院）；S.P. Timoshenko Institute of Mechanics of NAS of Ukraine（乌克兰国家科学院谢尔盖·P·蒂莫申科力学研究所）；Institute of Applied Mathematics and Mechanics of NAS of Ukraine（乌克兰国家科学院应用数学与力学研究所）
The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-Δ_p)^sv=0 \, , $$ where $$ (-Δ_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ has the interesting feature that an associated Rayleigh quotient is non-increasing in time along solutions. We prove the existence of a weak solution of the corresponding initial value problem which is also unique as a viscosity solution. Moreover, we provide Hölder estimates for viscosity solutions and relate the asymptotic behavior of solutions to the eigenvalue problem for the fractional $p$-Lap…
We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator, $$ \int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\, {\rm d}y, $$ where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$. Our results do also apply fo…
We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci.
The well-posedness of nonlocal elliptic equation with singular drift is investigated in Besov-Hölder spaces. As an application, we show the existence and uniqueness for corresponding martingale problem. Moreover, we prove that the one dimensional distribution of the martingale solution has a density in some Besov space.
We prove Hölder regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schrödinger equations subject to exterior Dirichlet conditions in an open set $Ω\subset \mathbb{R}^N$. The class of nonlocal operators considered here are defined, via Dirichlet forms, by kernels $K(x,y)$ bounded from above and below by $|x-y|^{N+2s}$, with $s\in (0,1)$. The entries in the equations are in some Morrey spaces and the underline domain $Ω$ satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When $K$ defines a nonlocal operator…
The aim of this article is to develop the regularity theory for parabolic equations driven by nonlocal operators associated with nonsymmetric forms. Hölder regularity and weak Harnack inequalities are proved using extensions of recently established nonlocal energy methods. We are able to connect the theory of nonsymmetric nonlocal operators with the important results of Aronson-Serrin in the local linear case. This connection is exemplified by nonlocal-to-local convergence results identifying the limiting class of operators as second order differential operators with drift terms.
We prove Hölder regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, the Hölder continuity of viscosity solutions is obtained. We get the similar results as those obtained by Allen (see {\url{https://arxiv.org/abs/1610.10073}})
In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-Δ_p)^s u&=& \fracλ{u^γ}+u^q~\text{in}~Ω,\nonumber u&=&0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber u&>& 0~\text{in}~Ω\nonumber, \end{eqnarray} where $Ω$ is an open bounded domain in $\mathbb{R}^N$ with smooth boundary, $N>ps$, $s\in (0,1)$, $λ>0$, $0<γ<1$, $1<p<\infty$, $p-1<q\leq p_s^{*}=\frac{Np}{N-ps}$. We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that f…
We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.
Hölder regularity for the parabolic perturbed fractional 1-Laplace equations
Harbin Institute of Technology（哈尔滨工业大学）；School of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
We establish sharp higher-order Hölder regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\timesΩ$ where $I\subset\mathbb{R}$, $Ω\subset\mathbb{R}^n$ and $f$ is Hölder continuous. The nonlocal operators $L$ considered are those arising in stochastic processes with jumps such as the fractional Laplacian $(-Δ)^s$, $s\in(0,1)$. Our main result establishes that, if $f$ is $C^γ$ is space and $C^{γ/2s}$ in time, and $Ω$ is a $C^{2,γ}$ domain, then $u/d^s$ is $C^{s+γ}$ up to the boundary in space and $u$ is $C^{1+γ/2s}$ up the boundary in time…
Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.
Well-posedness of stochastic time-nonlocal telegraph equations with Hölder diffusion coefficient: hereditary phase-space lifting and novel generalized coupling method
Faculty of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；Académico Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（奥希金斯大学工程科学研究所）；Faculty of Information Technology, Macau University of Science and Technology（澳门科技大学信息科技学院）
This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic in the sense of \cite{SS}, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.
