CONVERGENCE OF GENERALIZED ORLICZ NORMS WITH LOWER GROWTH RATE TENDING TO INFINITY

A BSTRACT . We study convergenceof generalized Orlicz energies when the lower growth-rate tends to inﬁnity. We generalize results by Bocea–Mih˘ailescu (Orlicz case) and Eleuteri–Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.


INTRODUCTION
In this paper we consider Γ-convergence of a sequence of energies whose growth-rates tend to infinity.Let us first recap motivation and earlier studies.Garroni, Nesi and Ponsiglione [16] studied dielectric breakdown using Γ-convergence of power-growth functions as the exponent tends to infinity.A more complicated functional of power-type was considered by Katzourakis [24] in a model for fluorescent optical tomography.Prinari and co-authors [10,28,29] considered an abstract version with the convergence of the energy as p → ∞.A similar problem without dependence on the derivative was considered in [1,6,7] and related to applications in polycrystal plasticity.If the energy is of non-standard growth-type [25], then the Γ-limit can contain singular parts [27].The energy I p has been generalized to two complementary cases: Orlicz growth by Bocea and Mihȃilescu [5] and variable exponent growth by Eleuteri and Prinari [15].In this article, we consider the result in generalized Orlicz spaces, which covers both of these as special cases.Generalized Orlicz spaces (also known as Musielak-Orlicz spaces) and related PDE have been intensely studied recently, see, e.g., [12,20,21,22,23] and references therein.Both the L ∞ -and the nonstandard growth energy have been related to image processing [9,11,19].
We follow the general approach of previous papers but we develop improved techniques with streamlined proofs and weaker assumptions.The tools used in [5,15] led to the unexpected assumption that ratios of the lower and upper growth-rates of the approximating functionals (denoted ϕ ± n and p ± n in these papers) are bounded.We develop more precise estimates which allow us to avoid this assumption.Indeed, we need no assumptions about the upper growth-rate and can consider non-doubling functionals as well.For instance, we can handle the energy |Du| n+1/|x| that was excluded from previous results.In addition, we are able to deal with open sets of finite measure, whereas previous results required regularity of the boundary.We cover both norm-and modulartype energies with much the same method (cf.Theorems 4.4).The general ϕ-growth highlights the exact properties that are needed for the results more clearly than the L p -case, as a comparison of the assumptions in Theorems 4.2 and 4.4 shows, see also Example 4.5.
We start by recalling some prior results related to Young measures and generalized Orlicz spaces.In Section 3, we prove key tools for transferring results for constant exponents to the generalized Orlicz case.We conclude in Section 4 with the main convergence results.

AUXILIARY RESULTS
Throughout the paper we always consider a domain Ω ⊂ R N , N 2, i.e. an open and connected set.We denote the set of measurable functions f : Ω → R d by L 0 (Ω, R d ).When d = 1, the second set is omitted and we write L 0 (Ω); the same convention is used for other function spaces.The Lebesgue and Borel measures on R k are denoted L k and B k , respectively, and The following is a substitute for Jensen's inequality for level convex functions.Lemma 2.2 (Theorem 1.2, [4]).Let f : R N → R be a lower semicontinuous and level convex function and let µ be a probability measure in the open set U ⊂ R N .Then Young measures.We recall some results regarding Young measures following the presentation of Section 2.3, [15] (which is based on [30]).
) is lower semicontinuous for a.e.x ∈ Ω.We refer to Corollary 2.10 in [15] for the next result.The measures µ x in the lemma are the so-called Young measures.
Lemma 2.4.Suppose that Ω is bounded and u n ⇀ u in W 1,q (Ω; R d ), q ∈ (1, ∞).Then there exists a family of probability measures (µ x ) x∈Ω on R N d such that Du(x) = ˆRNd ξ dµ x (ξ) for a.e.x ∈ Ω and, for any normal integrand f : With the notation of the previous lemma and v ∈ L 0 (Ω, R d ), [15,Remark 2.11] states that Combined with the inequality µ x -ess sup ξ∈R k f (x, v(x), ξ) f (x, v(x), Du(x)) from Lemma 2.2, this implies the following: Corollary 2.5.With the assumptions and notation of the previous lemma, we have provided that f (x, z, •) is level convex for every z ∈ R d and a.e.x ∈ Ω.
Generalized Orlicz spaces.In this subsection, we provide background on generalized Orlicz spaces.For more details, see [12,18].The next condition, which measures the lower growth-rate, can be stated as f (t) t p being almost increasing (hence the abbreviation).We use an equivalent form which is easier to apply.Definition 2.6.Let f : Ω × [0, ∞) → R and p > 0. We say that f satisfies (aInc) p if there exists L 1 such that f (x, λt) Lλ p f (x, t) for a.e.x ∈ Ω and all t 0, λ 1.
Note that if ϕ satisfies (aInc) p 1 , then it satisfies (aInc) p 2 , for every p 2 < p 1 .The condition (aDec) p , related to upper growth-rate, is defined by the same inequality when λ 1, but we do not need it in this paper.
We can now define generalized Orlicz spaces.Example 2.9 shows that this framework covers also L ∞ -spaces without the need for special cases.This is of special importance in this article as we consider the limit when the growth-rate tends to infinity.Definition 2.8.Let ϕ ∈ Φ w (Ω) and let the modular ̺ ϕ be given by is called a generalized Orlicz space.It is equipped with the Luxenburg quasinorm From the definition of modular we see that ∞, otherwise.

ASYMPTOTICALLY SHARP EMBEDDINGS
The unit ball property is a fundamental relation between quasinorm and modular.Note that we do not assume lower semi-continuity of ϕ which complicates the relationship is a bit.Lemma 3.1 (Unit ball property, Lemma 3.2.3,[18]).If ϕ ∈ Φ w (Ω), then This next property anchors ϕ at 1.It follows from (aInc) 1 that if 1 c ϕ(x, 1) c for a.e.x ∈ Ω, then ϕ satisfies (A0) with β := 1 Lc .Definition 3.2.We say that a function ϕ ∈ Φ w (Ω) satisfies (A0) if there exists β ∈ (0, 1] such that ϕ(x, β) 1 ϕ(x, 1 β ) for a.e.x ∈ Ω.The following proposition plays a significant role in reducing our main problem to the classic Lebesgue space L q (Ω), where we can use the results about Young measures previously presented.The embedding is well-known (see [18,Lemma 3.7.7])but we need a version with an asymptotically sharp constant as p → ∞.Then L ϕ (Ω) ֒→ L p (Ω) and Proof.The condition (A0) follows from 1 c ϕ(x, 1) c and (aInc) p , and so the embedding holds [18,Lemma 3.7.7].By the unit ball property (Lemma 3.1), the norm inequality u L p C u ϕ follows once we prove that Let us derive a lower bound for ϕ.By (aInc) p , ϕ(x, t) If we subtract 1 c from the right-hand side, we obtain a function which is negative when t < L 1/p and therefore a suitable lower bound also when t is in this range.It follows that ϕ(x, t) 1 Lc t p − 1 c for every t 0. Applying this in the modular, we see that Thus the desired norm-inequality holds if the right-hand side is greater than 1, which is equivalent to The next proposition essentially takes care of the lim sup-part of the estimate in the main results.The almost increasing assumption is satisfied (with L = 1) for example when ϕ • ϕ n satisfies (aInc) pn with constant L for some p n 1.
Proof.If u ∞ = 0, then u = 0 and there is nothing to prove.It suffices to consider the case u ∞ = 1, since the general case can be reduced to it by the normalization ũ := u u ∞ .By the definition of limit, we have to find for every ε ∈ (0, 1) a number n 0 ∈ N such that for all n n 0 .By the unit ball property (Lemma 3.1) this inequality follows from Lc(1 − ε) pn .Since |u| u ∞ = 1 a.e. and ϕ n is increasing, we conclude that Thus we can find n 0 such that ̺ ϕn ((1 − ε)u) < 1 for all n n 0 .
To deal with the second inequality, we consider The previous two results used the assumption 1 c ϕ n (x, 1) c which is stronger that (A0).This is because (A0) is not sufficient, as the next example demonstrates.
Example 3.5.Define ϕ n ∈ Φ w by ϕ n (t) := (a n t) n , where β a n Therefore, the conclusion of the previous proposition does not hold in this case.If a > 1, then the inequality u ϕn c n u ∞ of Proposition 3.3 similarly does not hold with any constant c n → 1.
The second example shows that the assumption that each ϕ n satisfy (aInc) pn with the same constant L is also needed for the conclusion.

MAIN THEOREMS
and otherwise.The abbreviations E ∞ and F ∞ refer to the case ϕ = ϕ ∞ from Example 2.9, related L ∞ .Remark 4.1.It follows from Theorems 4.2 and 4.4 that (F ϕn ) and (E ϕn ) Γ-converge to F ∞ and E ∞ respectively, with respect to the L 1 -weak topology.Specifically, we prove the lim inf-property of Γ-convergence and show that we can use a constant recovery sequence; we refer the reader to [8] for the definitions related to Γ-convergence which are not otherwise needed here.
Without loss of generality, we assume that Then every subsequence of (u n ) n also has limit M. Recall that p n → ∞.Fix q > γ and let n 0 ∈ N be such that p n q γ and F n (u n ) M + 1 for all n n 0 .
From p n q γ it follows that ϕ n satisfies (aInc) q/γ with the same constant L as in the assumption.By Proposition 3.3, for every n n 0 .We note that C q ց 1 as q → ∞.
We use this inequality and the assumed lower bound on f to estimate the norm of the gradient: ).It follows that ( Du n q ) n is bounded.Then, up to a subsequence (depending on q), (Du n ) n weakly converges to a function w in L q (Ω, R N d ).Since (u n ) n weakly converges to u in L 1 (Ω, R d ), w is the distributional gradient of u.Fix an open ball B ⊂ Ω.We use a version of the Poincaré inequality where L q -integrability is required only for the gradient.By the theorem in Section 1.5.2 [26, p. 35], we obtain that the right hand side is uniformly bounded.Thus reflexivity in L q (B, R d ) yields that (u n ) n has a weakly convergent subsequence.
We have found a subsequence, denoted still by (u n ) n , with u n ⇀ u in W 1,q (B, R d ).Let (µ x ) x∈B be the family of probability measures (the "Young measure") from Lemma 2.4 corresponding to this subsequence.We use the first inequality from (4.3) in B and Lemma 2.4 for the normal integrand f q/γ to conclude that .
Take the limit inferior on the right-hand side as q → ∞ and use C q → 1 as well as Corollary 2.5: where A ⊂ B is the exceptional set of measure zero related to the essential supremum.We can write Ω = ∪ i∈N B i as a countable union of balls B i and then we obtain an estimate for the supremum in , where A i is the exceptional set corresponding to B i .Since also ∪ i∈N A i has measure zero, this gives the essential supremum in Ω.Thus we have proved the lim inf-inequality and completed the proof.
Proof.We again abbreviate E n := E ϕn , n ∈ N. Let us first show that ) = 0 and the inequality follows.
To deal with the lim inf-inequality, let (u n ) n ⊂ L 1 (Ω, R d ) converge weakly to u ∈ L 1 (Ω, R d ).Without loss of generality, we assume that .
Since φn satisfies (aInc) pn with the same constant L, this yields by Corollary 3.  On the other hand, the energy ϕ n (t) := 1 n t n does satisfy the conditions the previous theorem.

Example 4 . 5 .
Let p n := n for all n.Then ϕ n (t) := t n satisfies (aInc) n and 1 c ϕ n (x, 1) c but not the condition lim sup n→∞ ϕ + n (1) = 0 of the previous theorem.Moreover, the claim of the theorem also does not hold: if f and u are such that f (x, u, Du) = 1 a.e., thenE n (u) = |Ω| for every n ∈ N but E ∞ (u) = 0 so that lim sup n→∞ E n (u) > E ∞ (u).