Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

Gwiazda, Piotr; Wittbold, Petra; Wróblewska, Aneta; Zimmermann, Aleksandra

doi:10.1016/j.jde.2012.03.025

Cited by 55 publications

(76 citation statements)

References 32 publications

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“…variable. We also use some idea about renormalized solutions for an elliptic problem [13,14]. Crucial tools in our consideration are: the L ∞ -truncation method and the result concerning the density of smooth function w.r.t.…”

Section: Previous Resultsmentioning

confidence: 99%

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Elliptic problems in generalized Orlicz-Musielak spaces

2012

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We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a ∆ 2 nor ∇ 2 -condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to . We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞ -truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces. MSC:46E30, 35J60

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Section: Previous Resultsmentioning

confidence: 99%

“…The conditions 2. and 4. of Definition 1.1 together with the uniform boundedness of { * (ε∇ (∇T ( ε )))} 1≥ε>0 in L 1 (Ω), see (13), provide the uniform integrability of {ε∇ (∇T ( ε ))} 1≥ε>0 We can estimate the second term in (21) as follows:…”

Section: − − → T ( ) Since = DIV F F ∈ E M *mentioning

confidence: 99%

Elliptic problems in generalized Orlicz-Musielak spaces

2012

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“…We use the framework developed in [32,18,37,38,39], where elliptic and parabolic problems in the Musielak-Orlicz spaces were studied, and apply the results of [17]. Since in general M * ∈ ∆ 2 , the understanding of the dual pairing is not intuitive.…”

Section: The Methodsmentioning

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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“…For this one can either on the basis of [50] follows the arguments of [21,29], or using considerations from [8,Section 7] and [24,15,29]. Analogical result on existence of renormalized solutions for problems with lower-order terms and L 1 -data (posed in Musielak-Orlicz space) is provided in [36].…”

Section: Notion Of Sola and Its Existencementioning

confidence: 99%

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

Chlebicka

2020

Isr. J. Math.

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Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the formwith data µ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function, whose speed of growth modulates the regularization of the solutions.

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Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces

Cited by 55 publications

References 32 publications

Elliptic problems in generalized Orlicz-Musielak spaces

Elliptic problems in generalized Orlicz-Musielak spaces

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

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

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