Existence and regularity of minimizers of nonconvex integrals with<i>p-q</i>growth

Celada, Pietro; Cupini, Giovanni; Guidorzi, Marcello

doi:10.1051/cocv:2007014

Cited by 19 publications

(12 citation statements)

References 24 publications

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“…The next theorem takes a first step towards the proof of the result |E i | = 0, by using some arguments presented by Dacorogna and Marcellini [35], Fonseca, Fusco and Marcellini [29] and Celada, Perrota and Guidorzi [45]. Theorem 4.3.…”

Section: Existence Theorem Of a Minimum For Each Functional Of The CLmentioning

confidence: 97%

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Reconstruction of noisy signals by minimization of non-convex functionals

Mederos

Mollineda

2016

Nonlinear Analysis: Real World Applications

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Non-convex functionals have shown sharper results in signal reconstruction as compared to convex ones, although the existence of a minimum has not been established in general. This paper addresses the study of a general class of either convex or non-convex functionals for denoising signals which combines two general terms for fitting and smoothing purposes, respectively. The first one measures how close a signal is to the original noisy signal. The second term aims at removing noise while preserving some expected characteristics in the true signal such as edges and fine details. A theoretical proof of the existence of a minimum for functionals of this class is presented. The main merit of this result is to show the existence of minimizer for a large family of non-convex functionals.

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Section: Existence Theorem Of a Minimum For Each Functional Of The CLmentioning

confidence: 97%

“…Following the strategy used to prove the equivalence of (44) and (45) in the Theorem 3.5 in [51], the next functional is proposed to establish an equivalence with M * * :…”

Section: Dealing With Discontinuities Via the Mumford-shah Functionalmentioning

confidence: 99%

Reconstruction of noisy signals by minimization of non-convex functionals

Mederos

Mollineda

2016

Nonlinear Analysis: Real World Applications

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“…In this case the natural idea, which is somehow common to [6,10,13,14], would be that of cutting away the degeneracy region, by localizing equation (1.7) "in a neighborhood of infinity". In a nutshell, this is done by selecting suitable test functions in the weak formulation of (1.7), for examples quantities like (|∇u 0 | β − M β 0 ) + would do the job.…”

Section: 2mentioning

confidence: 99%

“…More general functionals of this type have been considered for example in [10,13,14]. As pointed out in [7], regularity results for minimizers of such a kind of functionals are tightly connected with optimal transport problems with congestion effects.…”

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confidence: 99%

On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds

Brasco

Carlier

2013

Advances in Calculus of Variations

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Abstract. Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: ∂ x ( | u x | - δ 1 ) + q - 1 u x | u x | + ∂ y ( | u y | - δ 2 ) + q - 1 u y | u y | = f , $ \partial _x \biggl [(|u_{x}|-\delta _1)_+^{q-1}\, \frac{u_{x}}{|u_{x}|}\biggr ]+\partial _y \biggl [(|u_{y}|-\delta _2)_+^{q-1}\, \frac{u_{y}}{|u_{y}|}\biggr ]=f, $ for 2 ≤ q < ∞ ${2\le q<\infty }$ and some non-negative parameters δ 1 , δ 2 ${\delta _1,\delta _2}$ . Here ( · ) + ${(\,\cdot \,)_+}$ stands for the positive part. We prove that if f ∈ L loc ∞ ${f\in L^\infty _{\rm loc}}$ , then ∇ u ∈ L loc r ${\nabla u\in L^r_{\rm loc}}$ for every r ≥ 1.

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“…It is clear that if g = 0, then every function with gradient in K will satisfy the equation. Besides the papers cited before, we mention [1,5,9,17] where regularity issues were tackled and [6] where it is proven that solutions to (1.2) satisfy an obstacle problem for the gradient in the viscosity sense. Here, we will not specify the assumptions on A and g that guarantee the existence of a Lipschitz solution and we refer to the mentioned papers for this interesting issue.…”

Section: Introductionmentioning

confidence: 99%