We consider a Hilbertian and a charges approach to fractional gradient constraint problems of the type $|D^σu|\leq g$, involving the distributional fractional Riesz gradient $D^σ$, $0<σ<1$, extending previous results on the existence of solutions and Lagrange multipliers of these nonlocal problems.
We also prove their convergence as $σ\nearrow1$ towards their local counterparts with the gradient constraint $|D u|\leq g$.
Boundary value problems for nonlocal fractional elliptic equations with parameter in Banach spaces are studied. Uniform $L_p$-separability properties and sharp resolvent estimates are obtained for elliptic equations in terms of fractional derivatives. Particularly, it is proven that the fractional ellipitic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and …
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 show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
The maximal B_{p,q}^{s}-regularity properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal ellıpıtıc equation in B_{p,q}^{s} is sectorial and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces are established.
We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are established using approximating solutions and the theory of maximal monotone operators. Using the De Giorgi-Nash-Moser technique, we prove that the solutions are bounded and Hölder continuous for all positive time.
The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fract…
We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.
In this paper, for a degenerate higher order equation with a fractional derivative in the sense of Caputo, a nonlocal problem with conjugation conditions in a rectangular domain is studied. The solution is constructed in the form of a Fourier series in the eigenfunctions of a one-dimensional problem. A criterion for the uniqueness of a solution is given.
The aim of the present paper is to study existence results of minimizers of the critical fractional Sobolev constant on bounded domains. Under some values of the fractional parameter we show that the best constant is achieved. If moreover the underlying domain is a ball, we obtain positive radial minimizers for all possible values of the fractional parameter in higher dimension, while we impose a positive mass condition in low dimension.
In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.
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.
This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent $W^{s,p(.),\tilde{p}(.,.)}(Ω)$, when $p$ and $\tilde{p}$ satisfies some conditions. As application, we study the existence of solutions for a class of Kirchhoff-Choquard problem in $\mathbb{R}^N$.
This paper studies Langevin equation with nonlocal boundary conditions involving a $ψ$--Caputo fractional derivatives operator. By the aide of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, the $ψ$-fractional Gronwall inequality and $ψ$--fractional integration by parts are employed to prove Ulam--Hyers and Ulam--Hyers--Rassias stability for the solutions. Examples are gifted to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literat…
In this paper, we consider a diffusion equation with fractional-time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. Existence and uniqueness of solution are proved by means of a spectral argument. The existence of solution is obtained for all values of the fractional parameter $α\in (0,1)$. Moreover, by applying control theory to the fractional diffusion problem we obtain an optimality system which has also a unique solution.
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{…
In this work we introduce volume constraint problems involving the nonlocal operator $(-Δ)_δ^{s}$, closely related to the fractional Laplacian $(-Δ)^{s}$, and depending upon a parameter $δ>0$ called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when $δ\to0^+$ and $δ\to+\infty$. Through these limit processes on $(-Δ)_δ^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $δ\to 0^+$ and $δ\to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to sol…
In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces $W^{s,p}$ to include the general case $W^sL_A$, where $A$ is an N-function and $s\in (0,1)$. We are concerned with some qualitative properties of the space $W^sL_A$ (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces.
This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedn…
We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.
We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of $s$-fractional mean curvature flows as $s\to 0^+$ and $s\to 1^-$. In analogy with the $s$-fractional mean curvature flows, we introduce the notion of $s$-Riesz curvature flows and characterize its limit as $s\to 0^-$. Eventually, we discuss the limit behavior as $r\to 0^+$ of the flow generated by a regularization of the $r$…
This paper deals with the existence and uniqueness of solutions for a nonlinear boundary value problem involving a sequential $ψ$-Hilfer fractional integro-differential equations with nonlocal boundary conditions. The existence and uniqueness of solutions are established for the considered problem by using the Banach contraction principle, Sadovski's fixed point theorem, and Krasnoselskii-Schaefer fixed point theorem due to Burton and Kirk. In addition, the Ulam-Hyers stability of solutions is discussed. Finally, the obtained results are illustrated by examples.
We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L 1 initial data, general self-adjoint pure jump L{é}vy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the nonlocal dissipation measure. This approach is inspired from the second order theory unlike the cutting technique previously introduced for bounded entropy solutions. The latter technique no longer seems to fi…
We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these …
This paper contain a new discussion for the type of generalized nonlinear Caputo fractional $q$-difference equations with $m$-point boundary value problem and Riemann-Stieltjes integral $\tildeα[x]:=\int_{0}^{1}~x(t)dΛ(t).$ By applying the fixed point theorem in cones, we investigate an existence of a unique positive solution depends on $λ>0.$ We present some useful properties related to the Green's function for $m-$point boundary value problem.
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) $s$-minimal sets. We study the $s$-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter $s$. We introduce a functional framework for studying those $s$-minimal sets that can be globally written…
In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal $p(x,.)$-Kirchhoff type problem. Further…
In this paper, we establish a generalization of Sturm--Picone comparison theorem for a pair of fractional nonlocal equations: \begin{eqnarray*} \begin{gathered} (-div. (A_1(x)\nabla))^{s} u = C_{1}(x) u \,\,\,\mbox{in}\,\,Ω, u = 0 \,\,\,\,\mbox{on}\,\,\,\,\,\,\,\partial Ω, \end{gathered} \end{eqnarray*} and \begin{eqnarray*} \begin{gathered} (-div. (A_2(x)\nabla))^{s} v = C_{2}(x) v \,\,\,\mbox{in}\,\,Ω, v = 0 \,\,\,\,\mbox{on}\,\,\,\,\,\,\,\partial Ω, \end{gathered} \end{eqnarray*} where $Ω\subset \mathbb{R}^n$ is an open bounded subset with smooth boundary, $0<s<1,\,\,A_1,\,A_2$ are real sym…
In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$ (D_{K,A}) \hspace*{0.5cm} \left\{ \begin{array}{clclc} M\left( \displaystyle \int_{\R^{2N}}A\left( [u(x)-u(y)] K(x,y)\right) dxdy\right) \mathcal{L}^K_A u & = & f(x,u) & \text{ in }& Ω, \hspace*{7cm} u & = & 0 \hspace*{0.2cm} \hspace*{0.2cm} & \text{ in } & \R^N\setminus Ω. \label{eq1} \end{array} \right. $$ } Where $\mathcal{L}^K_A$ is a nonlocal operator with sing…
We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $σ$-gradient ($0<σ<1$). We establish continuous dependence results with respect to the data, including the threshold of the fractional $σ$-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Usin…
