Lipschitz continuity for energy integrals with variable exponents

Eleuteri, Michela; Marcellini, Paolo; Mascolo, Elvira

doi:10.4171/rlm/723

Cited by 53 publications

(38 citation statements)

References 17 publications

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“…is considered by the authors in [13]. Assumption 3.1 Assume that f ∈ C 2 ( × R N n ) and there exist two positive constants k and K such that for ξ ∈ R N n and a.e.…”

Section: Letmentioning

confidence: 99%

Lipschitz estimates for systems with ellipticity conditions at infinity

Eleuteri

Marcellini

Mascolo

2015

Annali di Matematica

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In the general vector-valued case N≥ 1 , we prove the Lipschitz continuity of local minimizers to some integrals of the calculus of variations of the form ∫Ωg(x,|Du|)dx, with p, q-growth conditions only for | Du| → + ∞ and without further structure conditions on the integrand g= g(x, | Du|). We apply the regularity results to weak solutions to nonlinear elliptic systems of the form ∑i=1n∂∂xiaiα(x,Du)=0, α= 1 , 2 , … , N

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“…is considered by the authors in [13]. Assumption 3.1 Assume that f ∈ C 2 ( × R N n ) and there exist two positive constants k and K such that for ξ ∈ R N n and a.e.…”

Section: Letmentioning

confidence: 99%

Lipschitz estimates for systems with ellipticity conditions at infinity

Eleuteri

Marcellini

Mascolo

2015

Annali di Matematica

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“…For the continuous variable exponent case, nowadays many results on the regularity for minimizer are known, see [21,22,23,24]. Further results in this direction can be, for instance, found in [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36,37,38,39,40,41] for partial regularity results for p(x)-energy type functionals:…”

Section: Introduction and Main Theoremmentioning

confidence: 99%

Regularity for minimizers for functionals of double phase with variable exponents

Ragusa

Tachikawa

2019

Advances in Nonlinear Analysis

269

112

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“…Then for every x ∈ Ω we can choose a cube Q ν(µ) j including x. Then, using (24) and (M), we realize that for arbitrary t ∈ I 1 µ i we get…”

Section: Approximation In Musielak-orlicz Spacesmentioning

confidence: 99%

“…This direction comes from the fundamental papers [51,52] by Marcellini and despite it is well understood area it is still an active field especially from the point of view of modern calculus of variations and potential theory, see e.g. [26,41,24,25,4,19,2,44].However, there is a vast range of N -functions that do not satisfy the ∆ 2 condition, e.g.• M (t, x, ξ) = a(t, x) (exp(|ξ|) − 1 + |ξ|);• M (t, x, ξ) = a(t, x)|ξ 1 | p1(t,x) (1 + | log |ξ||) + exp(|ξ 2 | p2(t,x) ) − 1, when (ξ 1 , ξ 2 ) ∈ R 2 and p i :. This is a model example to imagine what we mean by an anisotropic modular function.Resigning from growth restrictions requires some density properties of the space, cf.…”

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confidence: 99%

Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon

Chlebicka

Gwiazda

Zatorska–Goldstein

2019

Journal of Differential Equations

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We provide existence and uniqueness of renomalized solutions to a general nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N . Namely we studyThe growth of the monotone vector field A is assumed to be controlled by a generalized nonhomogeneous and anisotropic N -Existence and uniqueness of renormalized solutions are proven in absence of Lavrentiev's phenomenon. The condition we impose to ensure approximation properties of the space is a certain type of balance of interplay between the behaviour of M for large |ξ| and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (changing in time).The noticeable challenge of this paper is considering the problem in non-reflexive and inhomogeneous fully anisotropic space that changes along time. New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. The problems similar to (7) with A depending on ∇u only and with polynomial growth are very well understood. There are countless deep results concerning the corresponding problems involving the p-Laplace operator, A(t, x, ξ) = |ξ| p−2 ξ, stated in the Lebesgue space setting (the modular function is then M (t, x, ξ) = |ξ| p ). There is a wide range of directions in which the polynomial growth case has been developed, including the variable exponent, Orlicz, and double-phase spaces unified. Survey [15] describes how they can be unified in the framework of Musielak-Orlicz spaces and used as a setting for differential equations.The study of nonlinear boundary value problems in non-reflexive Orlicz-Sobolev-type setting originated in the work of Donaldson [22] and Gossez [28,29,30]. We refer to the paper of Mustonen and Tienari [55] for a summary of the results. The case of vector Orlicz spaces with an anisotropic modular function, but independent of spacial or time variables, was investigated in [35].The Musielak-Orlicz setting in full generality has been studied systematically starting from [54, 59, 60] and developed inter alia around the theory arising from fluid mechanics [33,34,36,62]. For other recent developments of the framework of the spaces let us refer e.g. to [1,41,42,43,49,50]. Typically the research concentrates, however, mostly on the ∆ 2 /∇ 2 -case, or -even if without structural conditions of ∆ 2 -type (and thus done in nonreflexive spaces) -when the modular function was trapped between some power-type functions usually briefly described as p, q-growth. This direction comes from the fundamental papers [51,52] by Marcellini and despite it is well understood area it is still an active field especially from the point of view of modern calculus of variations and potential theory, see e.g. [26,41,24,25,4,19,2,44].However, there is a vast range of N -functions that do not satisfy the ∆ 2 condition, e.g.• M (t, x, ξ) = a(t, x) (exp(|ξ|) − 1 + |ξ|);• M (t, x, ξ) = a(t, x)|ξ 1 | p1(...

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Lipschitz continuity for energy integrals with variable exponents

Cited by 53 publications

References 17 publications

Lipschitz estimates for systems with ellipticity conditions at infinity

Lipschitz estimates for systems with ellipticity conditions at infinity

Regularity for minimizers for functionals of double phase with variable exponents

Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak–Orlicz spaces in absence of Lavrentiev's phenomenon

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