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

Abstract

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(...

On renormalized solutions to elliptic inclusions with nonstandard growth

“…REMARK 3.1. The condition (C2) comes from [25,26] (also see [1] for the isotropic version) and it guarantees the modular density of smooth functions in the Musielak-Orlicz-Sobolev space (or, in other words, it excludes the so called Lavrentiev phenomenon [52]). If in addition we assume that M (x, ξ) ≥ c gr |ξ| p with p > 1 and c gr > 0, then (3.1) in (C2) can be replaced with the condition…”

“…We require that either the complementary function M * of the N -function M satisfies the ∆ 2 condition or the condition which we name (C2) holds. This condition (C2) coming from [25,26], guaranteeing the modular density of smooth functions, always holds when the N function is independent of x variable. So, in such case, i.e.…”

A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces

“…Chlebicka, Gwiazda, Zatorska-Goldstein and co-authors [1,6,7,8,9,10,11,16] considered the assumption (M) in the anisotropic case; in the next definition their condition is reformulated to make it easier to compare with the (A1) condition (see also Lemma 3.4); also note that some of the earlier works included additional restrictions in the condition. Definition 3.2.…”

“…To this end, we developed in the isotropic case the (A1) condition [17,21] (see also [27]), which is essentially optimal for the boundedness of the maximal operator. In the anisotropic case Chlebicka, Gwiazda, Zatorska-Goldstein and co-authors [1,6,7,8,9,10,11,16] have developed a theory based on their (M) condition. To state and compare these conditions, let us define In essence, the (A1) conditions says that Φ + B can be bounded by Φ − B in small balls B ⊂ R n in a quantitative way, whereas (M) say that it can be similarly bounded by the least convex minorant (Φ − B ) conv of Φ − B (see Definition 3.2).…”

On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals

“…Therefore, to control mollifications, we need a different method to approximate ψ(x, ξ) with a function depending only on ξ. The construction below is somehow standard and has appeared in many works before, see [11,12].…”