Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

Abstract

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consideron a Lipschitz bounded domain in R N . The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N -function M . The approach does not require any particular type of growth condition of M or its conjugate M * (neither ∆ 2 , nor ∇ 2 ). The condition we impose is log-Hölder continuity of M , which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.

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

“…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.…”

“…In the case of classical Orlicz spaces, the crucial density result was provided by Gossez [30]. In the case of xdependent and anisotropic log-Hölder continuous modular functions the absence of Lavrentiev's phenomenon was proven in [32,18], further refined in isotropic case in [1] to cover both -log-Hölder condition in the variable exponent and closeness of parameters in double-phase space sharp due to [19]. Finally, in [17] spaceapproximation and easy time-approximation results we need here are provided under our anisotropic conditions.…”

“…As a consequence of weak monotonicity, we will be able to identify some limits using the following monotonicity trick applied e.g. in [32,18,37,62]. Lemma 2.1 (Lemma 6.5, [17]).…”

“…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. [1,32,18], which we discuss in further parts of the paper. Particularly challenging is admitting not only space inhomogeneity, but also time dependence of the modular function.…”

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

“…Motivated by this, the study for the real-variable theory of various function spaces on R n or domains in R n , especially, the Hardy-type spaces, associated with different differential operators, has inspired great interests in recent years; see, for example, [3,4,31,32,34,46,49,52,73,78] for the case of Hardy spaces, [11,61,72] for the case of weighted Hardy spaces, [1,88,89,90] for the case of variable exponent Hardy spaces and [2,50,51,80,82,83,85] for the case of (Musielak-)Orlicz Hardy spaces. Recall that the Musielak-Orlicz space was originated by Nakano [67] and developed by Musielak and Orlicz [64,65], which is a natural generalization of many important spaces such as (weighted) Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces and not only has its own interest, but is also very useful in partial differential equations [6,7,44,40], in calculus of variations [27], in image restoration [43,54] and in fluid dynamics [77,62]. The Musielak-Orlicz Hardy space on R n has proved useful in harmonic analysis (see, for example, [56,18,57,79]) and, especially, naturally appears in the endpoint estimate for both the div-curl lemma and the commutator of Cald...…”