On renormalized solutions to elliptic inclusions with nonstandard growth

Abstract

We study the elliptic inclusion given in the following divergence formAs we assume that f ∈ L 1 (Ω), the solutions to the above problem are understood in the renormalized sense. We also assume nonstandard, possibly nonpolynomial, heterogeneous and anisotropic growth and coercivity conditions on the maximally monotone multifunction A which necessitates the use of the nonseparable and nonreflexive Musielak-Orlicz spaces. We prove the existence and uniqueness of the renormalized solution as well as, under additional assumptions on the problem data, its relation to the weak solution. The key difficulty, the lack of a Carathéodory selection of the maximally monotone multifunction is overcome with the use of the Minty transform. CONTENTS 1. Introduction 2. N -functions and Musielak-Orlicz spaces 3. Main results 4. Multivalued term and its regularization 5. Proof of Theorem 3.6: existence. 6. Proof of Theorem 3.7: uniqueness. 7. Proof of Theorem 3.12: relation between renormalized and weak solution. References Appendix

Measure data elliptic problems with generalized Orlicz growth

“…Indeed, then for each fixed k > 0, there exists a subsequence of {u s }, still indexed by s, such that 1) also (40) holds, that is we have (35) and (36). Due to Remark 13 we get uniform integrability of { (x, ∇T k u)} k , so Lebesgue's monotone convergence theorem justifies (38), where the limit is in (L 1 (Ω)) n by Lemma 18. By ( 29) and Vitali's convergence theorem we infer (37).…”

“…Given an interest one may expect developing our main goals further towards anisotropic or non-reflexive settings cf. [5,23,46], as well as by involving lower-order terms in (1) as in [45], differential inclusions as in [38], or systems of equations.…”