Elliptic problems in generalized Orlicz-Musielak spaces

Gwiazda, Piotr; Minakowski, Piotr; Wróblewska-Kamińska, Aneta

doi:10.2478/s11533-012-0126-3

Cited by 23 publications

(42 citation statements)

References 15 publications

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“…An analogous result in the case of standard procedure, namely division for star-shaped domains and only x−dependent modulars was presented by Benkirane et al [3], see also [21] for the extension to an anisotropic case.…”

Section: Preliminariessupporting

confidence: 67%

Nonlinear parabolic problems in Musielak–Orlicz spaces

Świerczewska-Gwiazda

2014

Nonlinear Analysis: Theory, Methods & Applications

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Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent N-function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic Hölder regularity with respect to t and x.

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Section: Preliminariessupporting

confidence: 67%

Nonlinear parabolic problems in Musielak–Orlicz spaces

Świerczewska-Gwiazda

2014

Nonlinear Analysis: Theory, Methods & Applications

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“…The weak lower semi-continuity of a convex functional together with the above a priori estimates imply existence of u ∈ V M T (Ω) such that (29), (30), (31) hold, and existence of A k such that (32) holds. A(t, x, ∇u n )∇u n dx dt = 0.…”

Section: Convergence Of Truncationsmentioning

confidence: 90%

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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“…To the best of authors' knowledge only the result from [4] concerns the existence of weak solutions of elliptic problems in which the growth condition is given by an anisotropic inhomogeneous N -function.…”

Section: Existence Of Solutions To Elliptic Problemsmentioning

confidence: 99%

Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces

et al. 2019

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We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic N -function M , which may also depend on the spatial variable, i.e., the homogenization process will change the underlying function spaces and the nonlinear elliptic operator at each step. The problem of homogenization of nonlinear elliptic systems has been solved for the L p −setting with restrictions either on constant exponent or variable exponent that is assumed to be additionally log-Hölder continuous. These results correspond to a very particular case of N -functions satisfying both ∆ 2 and ∇ 2 -conditions. We show that for general M satisfying a condition of log-Hölder type continuity, one can provide a rather general theory without any assumption on the validity of neither ∆ 2 nor ∇ 2 -conditions.

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Elliptic problems in generalized Orlicz-Musielak spaces

Cited by 23 publications

References 15 publications

Nonlinear parabolic problems in Musielak–Orlicz spaces

Nonlinear parabolic problems in Musielak–Orlicz spaces

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

Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces

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