Regularity for scalar integrals without structure conditions

Eleuteri, Michela; Marcellini, Paolo; Mascolo, Elvira

doi:10.1515/acv-2017-0037

Cited by 72 publications

(54 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof follows with a similar argument as in [13] (see also [15]). We are ready to prove the main result of the paper.…”

mentioning

confidence: 71%

“…The x−dependence cannot be only considered as a perturbation, but it is a relevant difference with the case f = f (ξ) from several points of view. For instance recently several authors studied the x−dependence under Hölder continuity assumptions as well as under Sobolev summability assumptions, in the general context of p, q−growth conditions; see [7], [8], [13], [14]; see also [11], [3], [4], [15]. In this paper we show that we can obtain the local Lipschitz continuity in Ω of the local minimizers by assuming a mild condition on the x−dependence, weaker than (4).…”

mentioning

confidence: 99%

“…A motivation for the previous result, other that its intrinsic interest in the framework of regularity theory, also relies in the approximation procedure to pass from a-priori estimates to existence and regularity under general p, q−growth conditions. A place where this procedure has been used is the author's paper [15]; we plan to go back to this problem in the next future to explain with more details the use of Theorem 1 to get local Lipschitz continuity and regularity of solutions in a general context.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

Eleuteri¹,

Marcellini²,

Mascolo³

2019

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proof follows with a similar argument as in [13] (see also [15]). We are ready to prove the main result of the paper.…”

mentioning

confidence: 71%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

Eleuteri¹,

Marcellini²,

Mascolo³

2019

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

show abstract

“…in the last example being a (x) , b (x) ≥ 0 and a (x)+b (x) > 0. The last one recently received large consideration in the mathematical literature, with the name double phase problem, in particular by Colombo-Mingione [25], [26], by Baroni-M.Colombo-Mingione [4], [5] and by Eleuteri-Marcellini-Mascolo [43], [44], [45].…”

mentioning

confidence: 99%

Regularity under general and <inline-formula><tex-math id="M1">\begin{document}$ p,q- $\end{document}</tex-math></inline-formula> growth conditions

Marcellini¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.…”

mentioning

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

View full text Add to dashboard Cite

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(...

show abstract

Regularity for scalar integrals without structure conditions

Abstract: AbstractIntegrals of the Calculus of Variations with {p,q}-growth may have not smooth minimizers, not even bounded, for general Show more

Cited by 72 publications

References 32 publications

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

Regularity under general and <inline-formula><tex-math id="M1">\begin{document}$ p,q- $\end{document}</tex-math></inline-formula> growth conditions

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

Contact Info

Product

Resources

About