Minimization of the Compliance under a Nonlocal p-Laplacian Constraint

Andrés, Fuensanta; Castaño, D.; Múñoz, Julio

doi:10.3390/math11071679

Cited by 3 publications

(1 citation statement)

References 46 publications

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“…Applying the maximum principle, they investigated the generalized time-fractional variable-order Caputo diffusion equations and fractional variable-order Riesz-Caputo diffusion equations. For other new developments of the maximum principle, the reader can refer to [8][9][10][11][12][13][14][15][16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

Guan,

Zhang

2023

Fractal Fract

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In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle.

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Section: Introductionmentioning

confidence: 99%

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

Guan,

Zhang

2023

Fractal Fract

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Stability of complement value problems for p-Lévy operators

Foghem

2024

Nonlinear Differ. Equ. Appl.

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We set up a general framework tailor-made to solve complement value problems governed by symmetric nonlinear nonlocal integro-differential p-Lévy operators. A prototypical example of integro-differential p-Lévy operators is the well-known fractional p-Laplace operator. Our main focus is on nonlinear integro-differential equations in the presence of Dirichlet, Neumann and Robin conditions and we show well-posedness results. Several results are new even for the fractional p-Laplace operator but we develop the approach for general translation-invariant nonlocal operators. We also bridge the gap from nonlocal to local, by showing that solutions to the local Dirichlet and Neumann boundary value problems associated with p-Laplacian are strong limits of the nonlocal ones.

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The Dual Approach to Optimal Control in the Coefficients of Nonlocal Nonlinear Diffusion

Schytt,

Evgrafov

2024

Appl Math Optim

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We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the $$p$$ p -Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem $$\Gamma $$ Γ -converges to its local counterpart, when the nonlocal horizon vanishes.

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Minimization of the Compliance under a Nonlocal p-Laplacian Constraint

Cited by 3 publications

References 46 publications

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

Stability of complement value problems for p-Lévy operators

The Dual Approach to Optimal Control in the Coefficients of Nonlocal Nonlinear Diffusion

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