Multiplicity of solutions for fractional $q(.)$-Laplacian equations

Rahmoune, Abita; Biccari, Umberto

doi:10.48550/arxiv.2103.12600

Cited by 4 publications

(6 citation statements)

References 37 publications

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“…Variable-order fractional Laplacian. In some recent contributions ( [7,10,79,95]), elliptic problems involving a variable-order fractional Laplacian (that is, with s = s(x) : Ω → (0, 1)) have been considered, analyzing the existence of (possible several) solutions and some optimal control issues applied to image denoising. It would be of interest to investigate parabolic problems (which,as far as we can tell, are still unaddressed) and associated theoretical and numerical control problems.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Control and numerical approximation of fractional diffusion equations

Biccari¹,

Warma²,

Zuazua³

2021

Preprint

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The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain Ω. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L 2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω ⊂ Ω. Secondly, we move our attention to the exterior control problem, in which the control region O ⊂ Ω c is located outside Ω. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.

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Section: Conclusion and Open Problemsmentioning

confidence: 99%

Control and numerical approximation of fractional diffusion equations

Biccari¹,

Warma²,

Zuazua³

2021

Preprint

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“…Their results were obtained using the MPT and Ekeland's variational principle. Finally, several open and interesting problems about the existence and multiplicity of solutions were highlighted at the end of their paper [47] (see of the aforementioned recent related works [41,[43][44][45][46] on the same subject).…”

Section: Introductionmentioning

confidence: 96%

“…In 2021, Rahmoune and Biccari [47] investigated a multiplicity of solutions for the fractional Laplacian operator involving variable exponent nonlinearities of the type…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of Solutions for Fractional-Order Differential Equations via the κ(x)-Laplacian Operator and the Genus Theory

Srivástava

Sousa

2022

Fractal Fract

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In this paper, we investigate the existence and multiplicity of solutions for a class of quasi-linear problems involving fractional differential equations in the χ-fractional space Hκ(x)γ,β;χ(Δ). Using the Genus Theory, the Concentration-Compactness Principle, and the Mountain Pass Theorem, we show that under certain suitable assumptions the considered problem has at least k pairs of non-trivial solutions.

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“…An important operator for posing fractional partial differential equations (FPDEs) is the fractional Laplacian, which has many equivalent definitions [7] although it is most commonly introduced using Fourier and inverse Fourier transforms. Detailed introductions and studies of the fractional Laplacian can be found in [8,9] and the references therein, while some related operators and extensions are studied for example in [10,11].…”

Section: Introductionmentioning

confidence: 99%

On the fractional Laplacian of a function with respect to another function

Fernandez,

Restrepo,

Djida

2024

Math Methods in App Sciences

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The theories of fractional Laplacians and of fractional calculus with respect to functions are combined to produce, for the first time, the concept of a fractional Laplacian with respect to a bijective function. The theory is developed both in the 1‐dimensional setting and in the general ‐dimensional setting. Fourier transforms with respect to functions are also defined, and the relationships between Fourier transforms, fractional Laplacians, and Marchaud‐type derivatives are explored. Function spaces for these operators are carefully defined, including weighted spaces and a new type of Schwartz space. The theory developed is then applied to construct solutions to some partial differential equations involving both fractional time derivatives and fractional Laplacians with respect to functions, with illustrative examples.

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Multiplicity of solutions for fractional $q(.)$-Laplacian equations

Cited by 4 publications

References 37 publications

Control and numerical approximation of fractional diffusion equations

Control and numerical approximation of fractional diffusion equations

Multiplicity of Solutions for Fractional-Order Differential Equations via the κ(x)-Laplacian Operator and the Genus Theory

On the fractional Laplacian of a function with respect to another function

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