Dirichlet’s principle and wellposedness of solutions for a nonlocal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo></mml:mrow></mml:math>Laplacian system

It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see [14]).

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“…But this amounts to say that u j is the solution of the minimization problem [25], or [16] for a general context). In particular…”

Section: A General Casementioning

confidence: 98%

Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

Andrés

Múñoz

2015

Journal of Mathematical Analysis and Applications

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“…Hence, the following identity D(|G(u)| p−2 G(u)) = L p u := 2 Ω ′ ∪Γ |u(y) − u(x)| p−2 (u(y) − u(x))µ(x, y)dy was also shown in [21, (5.3)] for p = 2. The general case was proved in [22], that is…”

Section: Network On G(x N K)mentioning

confidence: 98%

Nonlocal $p$-Laplacian Evolution Problems on Graphs

Hafiene¹,

Fadili²,

Elmoataz³

2018

SIAM J. Numer. Anal.

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In this paper we study numerical approximations of the evolution problem for the nonlocal p-Laplacian with homogeneous Neumann boundary conditions. First, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels and initial data). We then provide a similar bound for the case when one of the trajectories is discrete-in-time and the other is continuous. In turn, these results allow us to establish error estimates of the discretized p-Laplacian problem on graphs. More precisely, for networks on convergent graph sequences (simple and weighted graphs), we prove convergence and provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows. We finally touch on the limit as p → ∞ in these approximations and get uniform convergence results.

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“…The following result is a version of similar Poincaré's inequalities which may be found in [2,5,32,29].…”

Section: The Solution Space and A Nonlocal Poincaré's Inequalitymentioning

confidence: 80%

“…Finally, it is possible to provide a formula for integration by parts similarly to what was done in [3,32]. Note that, in contrast with the local theory, this nonlocal version does not contain any boundary term and thus the values on Γ could be nonzero without modifying the result.…”

Section: Nonlocal Vector Calculusmentioning

confidence: 96%

A Nonlocal Variational Model for Pansharpening Image Fusion

Duran¹,

Buades²,

Coll³

et al. 2014

SIAM J. Imaging Sci.

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Pansharpening refers to the fusion process of inferring a high-resolution multispectral image from a high-resolution panchromatic image and a low-resolution multispectral one. In this paper we propose a new variational method for pansharpening which incorporates a nonlocal regularization term and two fidelity terms, one describing the relation between the panchromatic image and the highresolution spectral channels and the other one preserving the colors from the low-resolution modality. The nonlocal term is based on the image self-similarity principle applied to the panchromatic image. The existence and uniqueness of minimizer for the described functional is proved in a suitable space of weighted integrable functions. Although quite successful in terms of relative error, state-of-theart pansharpening methods introduce relevant color artifacts. These spectral distortions can be significantly reduced by involving the image self-similarity. Extensive comparisons with state-ofthe-art algorithms are performed. 1. Introduction. Many earth resource satellites, such as IKONOS, Landsat, QuickBird, and SPOT, provide continuously growing quantities of remote sensing images useful for a wide range of both scientific and everyday tasks. For example, satellite images are used to improve visual photointerpretation [54], digital-surface model extraction [45], and texture analysis [48]. Further applications are feature detection [24], land cover classification [33], estimating water depth [38], soil moisture content [43], vegetation mapping [21], and many military tasks such as mission planning, navigation, and targeting.Digital color images are usually represented by three color values at each pixel. Nevertheless, most common cameras use a CCD sensor device measuring a single color per pixel. The other two color values of each pixel must be interpolated from the neighboring pixels in the so-called demosaicking process. The selected configuration of the CCD sensor usually follows the CFA Bayer, where, out of a group of four pixels, two are green, one is red, and one is blue. Most satellites use a different acquisition system that decouples the acquisition of a panchromatic image at high spatial resolution from the acquisition of a multispectral

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Dirichlet’s principle and wellposedness of solutions for a nonlocal p-Laplacian system

Cited by 29 publications

References 23 publications

Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

Nonlocal $p$-Laplacian Evolution Problems on Graphs

A Nonlocal Variational Model for Pansharpening Image Fusion

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