Variational problems of splitting-type with mixed linear-superlinear growth conditions
aa r X i v : . [ m a t h . A P ] J u l Variational problems of splitting-type withmixed linear- superlinear growth conditions
Michael Bildhauer & Martin Fuchs
Abstract
Variational problems of splitting-type with mixed linear- superlin-ear growth conditions are considered. In the twodimensional case theminimizing problem is given by J [ w ] = Z Ω h f (cid:0) ∂ w (cid:1) + f (cid:0) ∂ w (cid:1)i d x → minw.r.t. a suitable class of comparison functions. Here f is supposed tobe a convex energy density with linear growth, f is supposed to be ofsuperlinear growth, for instance to be given by a N -function or justbounded from below by a N -function. One motivation for this kindof problem located between the well known splitting-type problems ofsuperlinear growth and the splitting-type problems with linear growth(recently considered in [1]) is the link to mathematical problems inplasticity (compare [2]). Here we prove results on the appropriate wayof relaxation including approximation procedures, duality, existenceand uniqueness of solutions as well as some new higher integrabilityresults. In the last decades the study of variational problems with nonstandardgrowth conditions developed to one of the main topics in the calculus ofvariations and related areas. We do not want to go into details and willnot present the historical line of this development. The reader will find thisbackground information, for instance, in the recent paper [3].Let us just mention a few aspects, which serve as a motivation for themanuscript at hand. AMS-Classification: 49J45, 49N60
1s one of the first main contributions Giaquinta considered the most commonprototype of energies with ( p, q )-growth in the sense of minimizing a splittingfunctional Z Ω f ( ∇ u ) d x = Z Ω h f (cid:0) ∂ u (cid:1) + f ( ∂ u (cid:1)i d x → min (1.1)in a suitable class of comparison functions, where f and f are supposedto have different growth rates larger than 1. In [4] he presented a famouscounterexample which shows that in general and even in the scalar case wecannot expect the smoothness of solutions to this variational problem.Of course (1.1) also serves as a motivation to study variational problems withnon-uniform ellipticity conditions. As one variant we may consider energydensities of class C satisfying with different exponents 1 < p < q ( c , c > ξ , η ∈ R nN ) c (cid:0) | ξ | (cid:1) p − | η | ≤ D f ( ξ )( η, η ) ≤ c (cid:0) | ξ | (cid:1) q − | η | . (1.2)Here a lot of important contributions in the scalar and also in the vectorialsetting can be found, we just mention [5] as one central reference in the longseries of papers in this direction.Related to (1.2), Frehse and Seregin ([6]) considered plastic materials withlogarithmic hardening, i.e. the energy density f ( ξ ) = | ξ | ln (cid:0) | ξ | (cid:1) of nearlylinear growth. Due to [7] we have full regularity for this particular kind ofmodel.We finally pass to the case of linear growth problems for which the energydensity is of (uniform) linear growth w.r.t. the gradient but just satisfies anon-uniform ellipticity condition in the sense of (1.2). Of course the minimalsurface case is the most prominent representative for this kind of problemssatisfying with suitable constants a , b , c , c > a , b ≥
0, for all ξ , η ∈ R nN and with some exponent µ > a | ξ | − a ≤ f (cid:0) ξ (cid:1) ≤ b | ξ | + b ,c (cid:0) | ξ | (cid:1) − µ | η | ≤ D f ( ξ )( η, η ) ≤ c (cid:0) | ξ | (cid:1) − | η | . (1.3)In the minimal surface case we have µ = 3 and we like to mention the pioneer-ing work of Giaquinta, Modica and Souˇcek [8], [9] in the list of outstandingcontributions. In [8] and [9] a suitable relaxation is discussed together witha subsequent proof of apriori estimates. We note that the uniqueness of so-lutions in general is lost by passing to the relaxed problem.2n [10], condition (1.3) was introduced defining a class of µ -elliptic energydensities. The regularity theory for minimizers was studied in a series ofsubsequent papers, compare, e.g., [11]. We also like to mention the Lipschitzestimates of Marcellini and Papi [12], which cover a broad class of the func-tionals we discussed up to now.Very recently, the authors [1] considered variational problems of splitting-type as given in (1.1) but now with two energy parts being of linear growth.Here it turns out that the right-hand side of the ellipitcity condition in (1.3) isno longer valid and we just have for all ξ , η ∈ R nN with a positive constant cD f ( ξ )( η, η ) ≤ c | η | . Nevertheless, some natural assumptions still imply regularity and uniquenessproperties of solutions to the relaxed problem.In the manuscript at hand we follow this line of studying variational prob-lems of splitting-type by now considering variational problems with mixedlinear- superlinear growth conditions.A first step in this direction was already made in Chapter 6 of [13]. Theresults given there follow from suitable apriori estimates which are availableif the non-uniform ellipticty is not too bad. This leads to the analysis of theset of cluster points of minimizing sequences and to some kind of local inter-pretation for the stress tensor, although the existence and the uniqueness ofdual solutions were not established (compare Remark 6.15 of [13]).We like to finish this introductory remarks by mentioning a prominent appli-cation of a mixed linear- superlinear growth problem, which in [2] is discussedas the Hencky plasticity model. This problem takes the form (compare (4.17),Chapter I, of [2])inf v ∈C a ( Z Ω (div v ) d x + Z Ω ψ (cid:0) ε D ( v ) (cid:1) d x − L ( v ) o with a suitable class C a ⊂ W , (Ω; R ), some volume force L and the de-viatoric part ε D ( v ) of the symmetric gradient ε ( v ). Here div v enters withquadratic growth while the function ψ is of linear growth w.r.t. the tensor ε D ( v ). Let us note that this plasticity model is based on the dual point ofview, i.e. the so-called sur-potential ψ is introduced through the conjugatefunction ψ ∗ . In general a more explicit expression cannot be given (see Re-mark 4.1, p. 75 of [2]). One particular example is of the form ψ ( ξ D ) = Φ (cid:0) | ξ D | (cid:1) k , ν )Φ( s ) = νs if | s | ≤ k √ ν , √ k | s | − k ν if | s | ≥ k √ ν . However, at this stage our main difficulty in comparison to the known resultsfor the Hencky model is quite hidden. We postpone a refined discussion toRemark 2.1.Now let us introduce the general framework of our considerations in a moreprecise way. For the sake of notational simplicity we restrict our consid-erations to the case that a linear growth condition is satisfied in only onecoordinate direction.Suppose that Ω ⊂ R n is a bounded Lipschitz domain and let f : R n → R bean energy density of class C ( R n ) which is decomposed in the form f ( ξ ) = f ( ξ ) + f ( ξ ) , ξ = ( ξ , ξ ) ∈ R × R n − . (1.4)Here we assume that f : R → R and f : R n − → R are convex functionsof class C ( R ) and C ( R n − ), respectively, satisfying with a i , b i ≥ a , a , b > N -function A : R → R : a | ξ | − a ≤ f ( ξ ) ≤ a | ξ | + a , ξ ∈ R ,b A ( | ξ | ) − b ≤ f ( ξ ) , ξ ∈ R n − . (1.5)We wish to note that we work with energy densities of class C just for no-tational simplicity. This hypothesis just enters Section 4. It is easy to checkthat for example no differentiability assumptions are needed in Section 3.1,whereas the results of Section 3.2 hold for densities of class C .Having the decomposition (1.4) in mind, we now suppose n = 2 throughoutthe rest of this manuscript. This essentially clarifies the notation while themain ideas and results remain unchanged.For the definition and the properties of N -functions and Orlicz-Sobolev spacewe refer to the monographs [14] or [15]. The basics needed here are summa-rized in [16] or [17]. We suppose that A : [0 , ∞ ) → [0 , ∞ ) satisfies( N A is continuous, strictly increasing and convex.( N
2) lim t ↓ A ( t ) t = 0 and lim t →∞ A ( t ) t = ∞ . ( N
3) There exist constants k , t > A (2 t ) ≤ kA ( t )for all t ≥ t , N
3) is called a ∆ -condition near infinity. We note that there existsan exponent p ≥ c > c | t | p ≤ A ( t ) for all t ≫ . (1.6)Moreover we suppose some kind of triangle inequality for f : there exists areal number c > t , ˆ t ∈ R f (cid:0) t + ˆ t (cid:1) ≤ c h f (cid:0) t (cid:1) + f (cid:0) ˆ t (cid:1)i . (1.7)This condition, for instance, follows from the convexity of f together withsome ∆ -condition.For the definition of the Sobolev spaces W kp and their local variants we referto the textbook of Adams ([18]), the notation needed later in the case offunctions of bounded variation can be found, e.g., in the monographs [19]and [20]. For the sake of completeness we recall the definition of the Orlicz-Sobolev space generated by a N -function A satisfying ( N N
3) (see [14],[15]).In the following we suppose that the bounded Lipschitz domain Ω is nor-mal w.r.t. the x -axis (compare the approximation arguments presented inSection 2), i.e. there exist Lipschitz functions κ , κ : ( a, b ) → R such thatΩ = (cid:8) x ∈ R : x ∈ ( a, b ) , κ ( x ) < x < κ ( x ) (cid:9) . (1.8)The space L A (Ω) := ( u : Ω → R : u is a measurable function such thatthere exists λ > Z Ω A (cid:0) λ | u | (cid:1) d x < + ∞ ) is called Orlicz space equipped with the Luxemburg norm k u k L A (Ω) = inf ( l > Z Ω A | u | l ! d x ≤ ) . The Orlicz-Sobolev space is given by W A (Ω) = ( u : Ω → R : u is a measurable function , u, (cid:12)(cid:12) ∇ u (cid:12)(cid:12) ∈ L A (Ω) ) with norm k u k W A (Ω) = k u k L A (Ω) + k∇ u k L A (Ω) . W A (Ω) of C ∞ (Ω)-functions w.r.t. this norm is according toTheorem 2.1 of [16] (recall that we suppose ( N ◦ W A (Ω) = W A (Ω) ∩ ◦ W (Ω) . W ,A (Ω) := n w ∈ W , (Ω) : ∂ w ∈ L A (Ω) o ,E [ v ] := Z Ω f ( v ) d x , v ∈ L A (Ω) . In formal accordance with (1.9) we define the class ◦ W ,A (Ω) := W ,A (Ω) ∩ ◦ W (Ω) . observing that this set is just the completion of C ∞ (Ω) in W ,A (Ω) w.r.t. thenatural norm of W ,A (Ω).The main classes of functions under consideration are: C Sob := u + ◦ W ,A (Ω) , C BV := ( w ∈ BV(Ω) : k ∂ w k L A (Ω) < ∞ , ( w − u ) ν = 0 H -a.e. on ∂ Ω ) , where the boundary values u are always supposed to be of class W ∞ (Ω) (seealso Remark 6.12 in [13]). Note that in the definition of the space C BV werequire that the distributional derivative ∂ w is generated by a function fromthe space L A (Ω). Moreover, we consider the BV-trace of w and denote by ν = ( ν , ν ) the outward unit normal to ∂ Ω.With respect to these classes we consider the minimization problem J [ w ] := Z Ω f ( ∇ w ) d x → min in the class C Sob (1.10)and its relaxed version K [ w ] := Z Ω f (cid:0) ∂ a w (cid:1) d x + Z Ω f ∞ ∂ s w | ∂ s w | ! d | ∂ s w | + Z ∂ Ω f ∞ (cid:0) ( u − w ) ν (cid:1) d H + E (cid:2) ∂ w (cid:3) =: K [ w ] + E (cid:2) ∂ w (cid:3) → min in the class C BV . (1.11)6ere ∇ a w denotes the absolutely continuous part of ∇ w w.r.t. the Lebesguemeasure, ∇ s w represents the singular part.As a matter of fact, problem (1.10) in general is not solvable and one has topass to the relaxed version in order to have the existence of at least general-ized minimizers. The approach via relaxation in the case of linear growth iswell-known and outlined, e.g., in the monographs [19] or [20].It will turn out in Section 2 and in Section 3 that the functional K togetherwith the class C BV is the suitable choice in the setting at hand: in Section3.1 we show that there exists a solution u of problem (1.11). Moreover, withthe help of the geometric approximation procedure of Section 2, we showin Corollary 3.1 that the infima of (1.10) and (1.11) are equal. In the casethat the superlinear part is given by a N -function, we obtain in addition acomplete dual point of view.In Section 4 the apriori higher integrability and regularity results of the re-cent paper [1] on splitting-type variational problems with linear growth areessentially refined and carried over to the mixed linear- superlinear setting.Concerning the function f we suppose that there exist real numbers µ > γ ≥ c (cid:0) | t | (cid:1) − µ ≤ f ′′ ( t ) ≤ c (cid:0) | t | (cid:1) γ , t ∈ R , (1.12)holds with constants c , c >
0. We note that µ > f .For f we suppose that there exist real numbers ˆ µ < q ≥ c (cid:0) | t | (cid:1) − ˆ µ ≤ f ′′ ( t ) ≤ c (cid:0) | t | (cid:1) q − , t ∈ R , (1.13)holds with constants c , c >
0. Since f is of superlinear growth, the condi-tion ˆ µ < p , q )-growth of f wehave − ˆ µ = p − p >
1. We also note that for n > f : R n − → R such that withconstants λ , Λ > ξ , η ∈ R n − λ (cid:0) | ξ | (cid:1) − ˆ µ | η | ≤ D f ( ξ )( η, η ) ≤ Λ (cid:0) | ξ | (cid:1) q − | η | . (1.14)A Caccioppoli-type inequality w.r.t. ∂ ∇ u and ∂ ∇ u , respectively, using inaddition negative exponents gives different variants of regularity results de-pending on the properties of f , . 7n Section 5 we finally turn our attention to the question of uniqueness ofsolutions. First results were already given in Theorem 4.1 and Corollary 4.1by quoting [13].It remains to discuss the N -function case. If the ellipticity parameter from(1.12) satisfies µ <
2, then the smoothness properties of σ together with theuniqueness of σ imply the uniqueness of generalized solutions. In order tomake this argument precise we prove a generalization of [21], Theorem 7, tothe situation at hand. In this section we present an approximation procedure which is adapted tothe particular linear- superlinear setting. Although the arguments seem tobe quite technical, the principle idea is a geometric one.We have to take care of various aspects: • A retracting and smoothing procedure of the form u + η ε ∗ (cid:2) ( u − u )( x + δe ) (cid:3) is compatible with Lebesgue spaces. However it does not workw.r.t. the “BV”-direction e which is due to the possible concentrationof masses on the boundary. • In the linear growth situation the methods of local approximation (com-pare, e.g., [20], Theorem 1.17, p. 14) together with some extension by u outside of Ω (see, e.g., [8], [9]) serve as a powerful tool. However, apartition of the unity { ϕ i } is involved in this kind of argument. Thiscauses serious difficulties proving the convergence of f ( ∂ w m ) for theapproximating sequence w m since the derivatives of ϕ i do not cancelwhen calculating the integral of f evaluated at the corresponding ex-pression. • Combining and adjusting both methods and using the geometric struc-ture of the domain we obtain a partition of the unity such that thederivatives w.r.t. the relevant direction vanish.We start with a generalization of Lemma B.1 of [13] including strong L p -convergence of ∂ w m . The main new feature is the way of constructing thesequence { w m } which is crucial for proving Lemma 2.2. Lemma 2.1.
Let w ∈ BV(Ω) such that ∂ w ∈ L p (Ω) for some ≤ p < ∞ and such that ( w − u ) ν = 0 a.e. on ∂ Ω . hen there exits a sequence { w m } such that for all m ∈ N we have w m ∈ W (Ω) ∩ C ∞ (Ω) , ∂ w m ∈ L p (Ω) , trace w m = trace w and such that we theconvergences lim m →∞ Z Ω | w m − w | d x = 0 , lim m →∞ Z Ω p |∇ w m | d x = Z Ω p |∇ w | , lim m →∞ Z Ω | ∂ w m − ∂ w | p d x = 0 . Proof.
Recalling our assumption (1.8) imposed on the domain Ω we mayconsider w.l.o.g. the case Ω = ( − , × ( − , . (2.1)We fix a function w ∈ BV (Ω) and proceed in five steps. Step 1.
In the following we suppose that u = 0. The general case is ob-tained by considering w − u and adding u at the end of the proof.We then reduce the problem by choosing two smooth functions ψ , ψ :[ − , → [0 ,
1] such that ψ + ψ ≡ ψ ( t ) = 0 on [ − , − / ψ ( t ) = 1 on[1 / ,
1] and ψ ( t ) = 1 on [ − , − / ψ ( t ) = 0 on [1 / , ψ ( x ) w and ψ ( x ) w separately, hence w.l.o.g. w ≡ x = − Step 2.
Fix some ε > x -direction) w ε ( x ) = w ( x + ε e ) , (2.2)where w is extended by 0 on ( − , × [1 , ∞ ). At the end of our proof wepass to the limit ε → w ≡ h ( − , × (1 − ε , (cid:3) ∩ (cid:2) ( − , × ( − , − ε ) i . (2.3) Step 3.
We now take [20], proof of Theorem 1.17, as a reference (comparealso [13]), Lemma B.1), fix ε >
0, (recalling Step 1 and Step 2) and for l ∈ N we let Ω k = Ω lk := ( x ∈ Ω : − l + k < x < − l + k ) , k ∈ N , l is chosen sufficiently large such that Z Ω − Ω |∇ w | d x < ε . (2.4)With this notation we define A := Ω and A i = Ω i +1 − Ω i − := ( x ∈ Ω : − l + i + 1 < x < − l + i − − l + i − < x < − l + i + 1 ) =: (cid:8) x ∈ Ω : x ∈ I − i ∪ I + i (cid:9) . A partition { ϕ i } of the unity is defined w.r.t. these sets by ϕ i ∈ C ∞ ( A i ) , ≤ ϕ i ≤ , ∞ X i =1 ϕ i = 1 on Ω . For proving Lemma 2.2 below it will be crucial to observe that the functions ϕ i may be chosen respecting the structure of the stripes, i.e. for all i ∈ N ϕ i ( x , x ) = ˜ ϕ i ( x ) , ˜ ϕ i ∈ C ∞ (cid:0) I − i ∪ I + i (cid:1) . (2.5) Step 4.
Now we proceed essentially as described in Lemma B.1 of [13]: letΩ − = ∅ and denote by η a smoothing kernel. On account of (2.3) we select ε i small enough such that the smoothing procedure is well defined and suchthat we have spt η ε i ∗ ( ϕ i w ) ⊂ Ω i +2 − Ω i − , Z Ω (cid:12)(cid:12) η ε i ∗ ( ϕ i w ) − ϕ i w (cid:12)(cid:12) d x < − i ε , Z Ω (cid:12)(cid:12) η ε i ∗ ( w ∇ ϕ i ) − w ∇ ϕ i (cid:12)(cid:12) d x < − i ε , Z Ω (cid:12)(cid:12) η ε i ∗ ∂ ( ϕ i w ) − ∂ ( ϕ i w ) (cid:12)(cid:12) p d x < − i ε . (2.6)Moreover, the analogue to (2.3) holds for w m with some ˜ ε < ε . Here withthe choice ε = 1 /m we have set w m = ∞ X i =1 η ε i ∗ ( ϕ i w ) .
10y the above remarks we suppose with a slight abuse of notation (relabeling ε ) that we have in addition to (2.3) for all m ∈ N w m ≡ h ( − , × (1 − ε , (cid:3) ∩ (cid:2) ( − , × ( − , − ε ) i . (2.7)Given (2.7) we follow exactly the proof of Lemma B.1, where in particularthe notion of a convex function g of a measure (see [22]) is exploited via therepresentation Z U g ( ∇ w ) := sup κ ∈ C ∞ ( U : R n ) , | κ |≤ ( − Z U w div κ d x − Z U g ∗ ( κ ) d x ) and where g is of linear growth and g ∗ denotes the conjugate function (seethe definition given in Section 3.2). Step 5.
With ε ≪ ε (i.e. choosing m = m ( ε ) sufficiently large) we pass tothe limit ε →
0, which finally proves the lemma.Following the lines of Lemma 2.1 we obtain the convergence of the superlinearpart of the energy under consideration.
Lemma 2.2.
Given the notation of Lemma 2.1 we suppose that we have(1.5) and (1.6). Moreover, we now assume (1.7).Then the sequence { w m } of Lemma 2.1 satisfies Z Ω f ( ∂ w m ) d x → Z Ω f ( ∂w ) d x as m → ∞ . Proof.
We start with the first three steps of the proof of Lemma 2.1, inparticular we have (2.3), (2.7) and (2.5).If p is the exponent given in (1.6), then the strong L p -convergence of thesequence { ∂ w m } yields (after passing to a subsequence) ∂ w m → ∂ w a.e. in Ω. (2.8)The first ingredient of the proof follows from our assumption (1.7) andJensen’s inequality, where we recall that in fact only finite sums are con-sidered: for all x ∈ Ω and for all m ∈ N we have f (cid:0) ∂ w m (cid:1) = f ∂ ∞ X i =1 η ε i ∗ ( ϕ i w ) ! = f ∞ X i =1 η ε i ∗ ∂ ( ϕ i w ) ! ≤ c ∞ X i =1 f (cid:16) η ε i ∗ ∂ ( ϕ i w ) (cid:17) ≤ c ∞ X i =1 η ε i ∗ f (cid:0) ∂ ( ϕ i w ) (cid:1) . (2.9)11e also recall (2.5) which means ∂ ϕ i = 0. In conclusion, (2.9) shows f ( ∂ w m ) ≤ c ∞ X i =1 η ε i ∗ f (cid:0) ϕ i ∂ w ) . (2.10)Now we benefit from (2.8) and Egoroff’s theorem: for any ¯ ε > i ∈ N there exists a measurable set A i, ¯ ε such that | A i − A i, ¯ ε | < ¯ ε i ≪ ¯ ε and ∂ w m ⇒ ∂ w on A i, ¯ ε . (2.11)A suitable choice of ¯ ε i is made in (2.13).With the help of (2.10) one obtains for fixed i ∈ N (note that by the firstcondition of (2.6), there exist at most three different numbers k ∈ N suchthat the function η ε k ∗ f ( ϕ k ∂ w ) A i ) Z A i − A i, ¯ ε f ( ∂ w m ) d x ≤ c ∞ X k =1 Z A i − A i, ¯ ε η ε k ∗ f ( ϕ k ∂ w ) d x = c ∞ X k =1 Z A i − A i, ¯ ε " Z η ε k ( x − y ) f (cid:16)(cid:0) ϕ k ∂ w (cid:1) ( y ) (cid:17) d y d x = c ∞ X k =1 Z A i − A i, ¯ ε " Z B η ( z ) f (cid:16)(cid:0) ϕ k ∂ w (cid:1) ( x − ε k z ) (cid:17) d z d x ≤ c ∞ X k =1 Z B η ( z ) " Z T ki, ¯ ε f (cid:16)(cid:0) ∂ w (cid:1) ( y ) (cid:17) d y d z , (2.12)where it is abbreviated ( | U | denoting the Lebesgue measure of U ⊂ Ω) T ki, ¯ ε := (cid:8) y = x − ε k z : x ∈ A i − A i, ¯ ε (cid:9) , in particular | T ki, ¯ ε | = | A i − A i, ¯ ε | . Now, since for fixed i the sum is just taken over three indices, we may choose¯ ε i sufficiently small and finally obtain from (2.12) (recalling R Ω f ( ∂ w ) d x < ∞ ) Z A i − A i, ¯ ε f ( ∂ w m ) d x ≤ − i ¯ ε . (2.13)Decreasing ¯ ε i , if necessary, it may also be assumed that Z A i − A i, ¯ ε f ( ∂ w ) d x ≤ − i ¯ ε . (2.14)12y (2.3) and (2.7) we note once more that only finite sums have to be con-sidered and recalling (2.11), (2.13) and (2.14) we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Ω f ( ∂ w m ) d x − Z Ω f ( ∂ w ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N X i =1 Z A i (cid:12)(cid:12)(cid:12) f ( ∂ w m ) − f ( ∂ w ) (cid:12)(cid:12)(cid:12) d x ≤ N X i =0 Z A i, ¯ ε (cid:12)(cid:12)(cid:12) f ( ∂ w m ) − f ( ∂ w ) (cid:12)(cid:12)(cid:12) d x + 2¯ ε ≤ N X i =0 sup A i, ¯ ε (cid:12)(cid:12)(cid:12) f ( ∂ w m ) − f ( ∂ w ) (cid:12)(cid:12)(cid:12) | A i | + 2¯ ε ≤ ε provided that m > m with m sufficiently large. This finishes the proof ofthe lemma. Remark 2.1.
Now we can shortly discuss one main difference to the modelof Hencky plasticty investigated in [2]. There an approximation lemma isformulated as Theorem 5.3 in Chapter II. The convergence of the deviatoricpart in terms of the density f considered there corresponds to the convergenceof the square root in Lemma 2.1 which follows from the linear growth of f and the notion of a convex function of a measure.Our main difficulty is proving the convergence of the f -energy. In the caseof the Hencky plasticity the analogue is just a consequence of considering theintermediate topology (defined in formula (3.37), Chapter II, of [2]) whichrespects the linear operator div v , see also Theorem 3.4 and formula (5.53),Chapter II, of [2]. Now we define ˆΩ := ( − , × ( − , w ∈ BV(Ω) we letˆ w := (cid:26) w on Ω ,u on ˆΩ − Ω , where u represents a Lipschitz extension of our fixed boundary datum fromthe space W ∞ (Ω). 13e then have the validity of an approximation result corresponding to LemmaB.2 of [13]. It can be seen as a kind of generalization of Lemma 2.1 andLemma 2.2 where now C ∞ is replaced by C ∞ . Lemma 2.3.
Using the above notation suppose that w ∈ BV(Ω) , k ∂ w k L A (Ω) < ∞ and that we have (1.4), (1.5), (1.6) and (1.7). Then there exists a se-quence { w m } in u + C ∞ (Ω) such that passing to the limit m → ∞ we have i ) ˆ w m → ˆ w in L (cid:0) ˆΩ) ,ii ) Z ˆΩ p |∇ ˆ w m | d x → Z ˆΩ p |∇ ˆ w | ,iii ) Z Ω f ( ∂ w m ) d x → Z Ω f ( ∂ w ) d x . There are two approaches towards the existence of generalized solutions toproblem (1.10). The first one follows the direct method and leads to theexistence of solutions to problem (1.11). This works under quite weak as-sumptions, for example, the densities f and f need not to be of class C .The second approach yields the stress tensor as the unique solution of thedual problem and by the stress- strain relation a complete picture of thesituation is drawn. However, following the duality approach, we have tosuppose that f is given in terms of a N -function. Theorem 3.1.
Suppose that we have (1.4) - (1.7). Then the relaxed problem(1.11) admits a solution u ∈ C BV .Proof of Theorem 3.1. We recall that w.l.o.g. we suppose n = 2 and considera K -minimizing sequence (cid:8) u ( n ) (cid:9) in the admissible class C BV of comparisonfunctions. After passing to a subsequence we may assume that there exits afunction u ∈ BV(Ω) and a function v ∈ L p (Ω) such that as n → ∞ u ( n ) → u in L (Ω) , ∂ u ( n ) ⇁ v in L p (Ω) . (3.1)Here, in the case p = 1, we refer to the Theorem of De LaValee-Poussin (see,e.g., [15]).We have for any ϕ ∈ C ∞ (Ω) Z Ω u ( n ) ∂ ϕ d x = − Z Ω ∂ u ( n ) ϕ d x , v = ∂ u and since we have for any ψ ∈ C ∞ (Ω) Z Ω u ( n ) ∂ ψ d x = − Z Ω ∂ u ( n ) ψ d x + Z ∂ Ω u ψν d H , Z Ω u∂ ψ d x = − Z Ω ∂ uψ d x + Z ∂ Ω uψν d H , the convergences stated in (3.1) prove u ∈ C BV .We note thatlim inf n →∞ K [ u ( n ) ] ≥ lim inf n →∞ K (cid:2) u ( n ) (cid:3) + lim inf n →∞ E (cid:2) ∂ u ( n ) (cid:3) . By [23], see also [19], Theorem 5.47, p. 304, we have the lower semicontinuity K (cid:2) u (cid:3) ≤ lim inf n →∞ K (cid:2) u ( n ) (cid:3) . Discussing E we cite Theorem 2.3, p. 18, of [24], hence E (cid:2) ∂ u (cid:3) ≤ lim inf n →∞ E (cid:2) ∂ u ( n ) (cid:3) . Since (cid:8) u ( n ) (cid:9) was chosen as a K -minimizing sequence, the proof of Theorem3.1 is complete.Now, on account of our approximation Lemma 2.3, we have Corollary 3.1.
With the notation and under the hypotheses of Theorem 3.1we have inf w ∈C Sob J [ w ] = inf v ∈C BV K [ v ] = K [ u ] . Another approach leading to an analogue of the stress tensor, occurring asbasic quantity in problems from mechanics, is to consider the dual problem.As the main references on convex analysis we mention [25] and [26].Let us assume that we have (1.5) with A (cid:0) | ξ | (cid:1) = f ( ξ ) for all ξ ∈ R n − , with A being of class C (cid:0) [0 , ∞ ) (cid:1) and with A satisfying ( N N f : R → R , f ( ξ ) = f ( ξ ) + A (cid:0) | ξ | (cid:1) , ξ ∈ R . (3.2)As usual we define the conjugate function A ∗ : [0 , ∞ ) → [0 , ∞ ) by A ∗ ( s ) := max t ≥ (cid:8) st − A ( t ) (cid:9) . (3.3)15nd note that we have for all t ∈ [0 , ∞ ) A ( t ) + A ∗ (cid:0) A ′ ( t )) = tA ′ ( t ) . (3.4)In order to obtain a well-posed dual problem we additionally require A ∗ (cid:0) A ′ ( t ) (cid:1) ≤ c h A ( t ) + 1 i for all t ∈ R . (3.5)Since f ∗ ( s ) := sup ξ ∈ R n sξ − f ( ξ ) o , we obtain from the decomposition of ff ∗ ( ξ ) = f ∗ ( ξ ) + A ∗ ( | ξ | ) (3.6)as formula for the conjugate function f ∗ : R → R . The conjugate function f ∗ satisfies in correspondence to (3.4) f ( ξ ) + f ∗ (cid:0) Df ( ξ ) (cid:1) = ξ · Df ( ξ ) , ξ ∈ R . (3.7)Given these preliminaries we define the Lagrangian l ( v, τ ) := Z Ω τ · ∇ v d x − Z Ω f ∗ ( τ ) d x − Z Ω A ∗ (cid:0) | τ | (cid:1) d x ,v ∈ u + ◦ W ,A (Ω) , τ ∈ L ∞ ,A ∗ (Ω; R ) . (3.8)In (3.8) we have set L ∞ ,A ∗ (Ω; R ) := L ∞ (Ω) × L A ∗ (Ω) . With the help of the formula for the conjugate function given in (3.6) wehave the representation for the energy J defined in (1.10) J [ w ] = sup κ ∈ L ∞ ,A ∗ (Ω; R ) ( Z Ω κ · ∇ w d x − Z Ω f ∗ ( κ ) d x − Z Ω A ∗ (cid:0) | κ | (cid:1) d x ) , = sup κ ∈ L ∞ ,A ∗ (Ω; R ) l ( w, κ ) , w ∈ u + ◦ W ,A (Ω) . (3.9)The dual functional finally is defined via R [ τ ] := inf w ∈ u + ◦ W ,A (Ω) l ( w, τ ) , τ ∈ L ∞ ,A ∗ (Ω; R ) . (3.10)16his functional leads to the dual problem as the maximizing problem R [ τ ] → max in τ ∈ L ∞ ,A ∗ (Ω; R ) . (3.11)Then we have recalling Theorem 3.1 Theorem 3.2.
Suppose that we have our general assumptions (1.4) - (1.7).Moreover, suppose that f is given in (3.2) with A satisfying (3.5). Let u denote a solution of the problem (1.11).Then the “stress tensor” defined by σ ( x ) := Df (cid:0) ∇ a u (cid:1) = (cid:16) f ′ (cid:0) ∂ a u (cid:1) , A ′ (cid:0) | ∂ u | (cid:1)(cid:17) (3.12) is of class L ∞ ,A ∗ (Ω; R ) and maximizes the dual variational problem (3.11)with R given in (3.10).Proof. We first note that the boundedness of | f ′ | and condition (3.5) imply σ ∈ L ∞ ,A ∗ (Ω; R ).We then follow an Ansatz similar to Lemma 5.1 of [27]. For any v ∈ u + ◦ W ,A (Ω) we have recalling (3.8) und using (3.7) l ( v, σ ) = Z Ω ∇ v · Df (cid:0) ∇ a u (cid:1) d x − Z Ω f ∗ (cid:16) Df (cid:0) ∇ a u (cid:1)(cid:17) d x = Z Ω Df (cid:0) ∇ a u (cid:1) · (cid:0) ∇ v − ∇ a u (cid:1) d x + Z Ω f (cid:0) ∇ a u (cid:1) d x . (3.13)Now given | t | ≪ u t := u + t ( v − u ) ∈ u + ◦ W ,A (Ω). The K -minimalityof u obviously implies dd t | t =0 K [ u t ] = 0 , hence by ∇ s v = 00 = Z Ω Df (cid:0) ∇ a u (cid:1) · (cid:0) ∇ v − ∇ a u ) d x + dd t | t =0 Z Ω f ∞ ∂ s u t (cid:12)(cid:12) ∂ s u t (cid:12)(cid:12) ! d (cid:12)(cid:12) ∂ s u t (cid:12)(cid:12) + dd t | t =0 Z ∂ Ω f ∞ (cid:16)(cid:0) u − u t (cid:1) ν (cid:17) d H , (3.14)where ∂ s u t = (1 − t ) ∂ s u . Now we note thatdd t | t =0 Z Ω f ∞ ∂ s u (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) ! d (cid:16) (1 − t ) (cid:12)(cid:12) ∂ s u (cid:12)(cid:12)(cid:17) = − Z Ω f ∞ ∂ s u (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) ! d (cid:12)(cid:12) ∂ s u (cid:12)(cid:12) , v takes the boundary data u on ∂ Ω we havedd t | t =0 Z ∂ Ω f ∞ (cid:0) ( u − u t ) ν (cid:1) d H = − Z ∂ Ω f ∞ (cid:0) ( u − u ) ν (cid:1) d H . Hence, inserting (3.13) in (3.14) we have shown l ( v, σ ) = K [ u ] for any v ∈ u + ◦ W ,A (Ω)and taking the infimum w.r.t. the comparison function v we have R [ σ ] ≥ K [ u ] . (3.15)We already know from Section 2 that inf J = K [ u ] and the representation(3.9) finally yields ( w ∈ u + ◦ W ,A (Ω)) J [ w ] = sup κ ∈ L ∞ ,A ∗ (Ω; R ) l ( w, κ ) ≥ sup κ ∈ L ∞ ,A ∗ (Ω; R ) ( inf v ∈ u + ◦ W ,A (Ω) l ( v, κ ) ) = sup κ ∈ L ∞ ,A ∗ (Ω; R ) R [ κ ] , i.e.inf w ∈ u + ◦ W ,A (Ω) J [ w ] ≥ sup κ ∈ L ∞ ,A ∗ (Ω; R ) R [ κ ] . This together with (3.15) and Corollary 3.1 proves the theorem. q = 2 Let us start by recalling Theorem 6.5 of [13] together with our general as-sumption ˆ µ <
Theorem 4.1.
Suppose that we are given the assumptions (1.4), (1.5),(1.12) with µ < , γ = 0 and (1.13) with q = 2 . Let u ∗ ∈ M := n u ∈ BV(Ω) : u is the L -limit of a J -minimizingsequence from u + ◦ W (Ω) o . Then u ∗ is of class C ,α (Ω) for any < α < . Moreover, the elements of M are uniquely determined up to constants. emark 4.1. We again emphasize that the theorem holds without the re-striction n = 2 , i.e. the general case f : R n − → R λ (cid:0) | ξ | (cid:1) − ˆ µ | η | ≤ D f ( ξ )( η, η ) ≤ Λ | η | , ξ , η ∈ R n − ,λ , Λ > is included without being explicitely mentioned.Main idea of the proof of Theorem 4.1. The proof of the theorem is basedon uniform apriori estimates for the minimizers u δ of the standard quadraticregularization. However, is not immediate that { u δ } is a J -minimizing se-quence, if the superlinear part is not generated by a N -function.One way to overcome this difficulty is to introduce some kind of local regu-larized stresss tensor σ δ without knowing that a solution to a dual problemexists (see Remark 6.15 of [13]).This, together with the equation div σ δ = 0, leads to the minimality of weak L -cluster points of { u δ } via a generalized inf − sup relation. In conclusion, avariational inequality is derived for any u ∗ ∈ M which provides the regular-ity of u ∗ by Corollary 6.13 of [13] in the C ,α -regularity of the stress tensor.With our approximation result Corollary 3.1 we identify the elements of M with K -minimizers and observe that the class C BV is defined respecting thecondition ( w − u ) ν = 0 H a.e. on ∂ Ω. Corollary 4.1.
Given the assumptions of Theorem 4.1, the problem (1.11)is uniquely solvable.
With the help of Theorem 4.1 we now establish the first regularity result ofthis paper which is very much in the spirit of [1], i.e.: if we drop the elliptictycondition µ <
2, then we still have higher integrability of ∂ u . Theorem 4.2.
Suppose that we have the assumptions of Theorem 4.1 with-out the requirement µ < .Then there exists a generalized minimizer u ∈ M such that ∂ u ∈ L χ loc (Ω) for any finite χ . Proof of Theorem 4.2.
With the ideas presented in [1] in the linear growthcase, the main point is to introduce in the mixed linear-superlinear case asuitable regularization procedure to obtain a sufficiently smooth minimizingsequence. 19e fix 1 < ν < ν ( t ) := ( ν − Z t Z s (1 + r ) − ν d r d s , = t − − ν (1 + t ) − ν − ν − . Then Φ ν satisfies (1.12) with µ = ν and γ = 0 (in fact we may even choose γ = − f δ ( ξ ) = δ Φ ν ( | ξ | ) + f ( ξ ) + f ( ξ )we consider with the obvious meaning of notation the regularized minimiza-tion problem ( δ ≪ K δ [ w ] = K ,δ [ w ] + E (cid:2) ∂ w (cid:3) → min in the class C BV . (4.1)By Corollary 4.1 there exists a unique solution denoted by u δ and by Theo-rem 4.1 u δ is of class C ,α (Ω), hence arguing with the Euler equation we alsohave u δ ∈ W , loc (Ω).For the minimizing property of the sequence { u δ } we observe for any fixed w ∈ C Sob J [ u δ ] ≤ J δ [ u δ ] ≤ J δ [ w ] , hence lim inf δ → J [ u δ ] ≤ lim sup δ → J δ [ w ] = J [ w ] . Passing to a subsequence, if necessary, { u δ } obviously is a J -minimizing se-quence.Now we argue exactly as in [1], proof of Theorem 1, and complete the proofof Theorem 4.2. p -growth In this subsection we restrict our considerations to the case that the superlin-ear part is of p -power growth. This is done in order to simplify our expositionwhich otherwise would be based on additional parameters. Generalizationsare left to the reader. 20he main issue is that linear growth conditions are also compatible with p -growth conditions, p >
2. Higher integrability results for ∂ u are establishedin Theorem 4.3 under quite general assumptions.In Theorem 4.4 we suppose in addition that µ < p .Throughout this section we concentrate on the following model case:Given the general splitting hypothesis (1.4), we suppose that f is of lineargrowth, a | t | − a ≤ f ( t ) ≤ a | t | + a , t ∈ R ,a , a ≥ a , a >
0, satisfying for some µ > γ ≥ c (cid:0) | t | (cid:1) − µ ≤ f ′′ ( t ) ≤ ¯ c (cid:0) | t | (cid:1) γ , t ∈ R , (4.2)with constants c , ¯ c > γ > f is supposed to be of p -power growth, p >
1, in the sense that b | t | p − b ≤ f ( t ) ≤ b | t | p + b , t ∈ R ,b , b ≥ b , b >
0, satisfying in addition c (cid:0) | t | (cid:1) p − ≤ f ′′ ( t ) ≤ ¯ c (cid:0) | t | (cid:1) p − , t ∈ R , (4.3)with constants c , ¯ c > Theorem 4.3.
Suppose that the hypotheses of this section are valid.i) If γ = 0 , then there exists a generalized minimizer u ∈ M such that ∂ u ∈ L χ loc (Ω) for all finite χ > p . ii) If ≤ γ < pp + 1 , (4.4) then there exists a generalized minimizer u ∈ M such that ∂ u ∈ L χ loc (Ω) for some χ > p + 1 . roof of Theorem 4.3. The proof of Theorem 4.3 is carried out in four steps.
Step 1. Regularization.
We fix some 0 < δ < f δ ( ξ ) := δ (cid:0) | ξ | (cid:1) p + f ( ξ ) + f ( ξ ) , ξ = ( ξ , ξ ) ∈ R . We note that this Ansatz respects the splitting structure of the energy den-sity under consideration.We then consider the minimization problem J δ [ w ] := Z Ω f δ ( ∇ w ) d x → min in u + W ,p (Ω) (1.1 δ )with u δ denoting the unique solution of (1.1 δ ) satisfying in addition (see themonographs [28], Theorem 8.1, p. 267, and [29], Theorem 5.2, p. 277) u δ ∈ W , (Ω) ∩ W , ∞ loc (Ω) . (4.5)In the situation at hand one may use the dual problem in order to show thatthe sequence { u δ } is J -minimizing. To this purpose we let τ δ := ∇ f (cid:0) ∇ u δ (cid:1) ,σ δ := δX δ + τ δ = ∇ f δ (cid:0) ∇ u δ (cid:1) , X δ := p (cid:0) | ∂ u δ | (cid:1) p − ∂ u δ , and adapt the arguments of [13], Section 4.1.2: we note that σ δ is of class W , (recall (4.5)) satisfying div σ δ = 0 and by J δ [ u δ ] ≤ J δ [ u ] ≤ J [ u ]we have with a finite constant c independent of δ and not relabelled in thefollowing δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x ≤ c , i.e. k δ p − p X δ k p/ ( p − ≤ c (4.6) Z Ω f ( ∇ u δ ) d x ≤ c , k τ δ k p/ ( p − ≤ c . (4.7)Observe that (4.6) implies as δ → δX δ ⇁ L pp − (Ω) . (4.8)After passing to subsequences we obtain from (4.7) and (4.8) as δ → τ δ , σ δ ⇁ : σ in L pp − (Ω) . (4.9)We recall the equation τ δ : ∇ u δ − f ∗ ( τ δ ) = f ( ∇ u δ )22nd arrive at J δ [ u δ ] = δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x + Z Ω (cid:2) τ δ : ∇ u δ − f ∗ ( τ δ ) (cid:3) d x = δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x + Z Ω (cid:2) σ δ : ∇ u δ − f ∗ ( τ δ ) (cid:3) d x − δp Z Ω (cid:0) | ∂ u δ | (cid:1) p − |∇ u δ | d x Using div σ δ = 0 we obtain J δ [ u δ ] = δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x + Z Ω (cid:2) σ δ : ∇ u − f ∗ ( τ δ ) (cid:3) d x − δp Z Ω (cid:0) | ∂ u δ | (cid:1) p − |∇ u δ | d x = Z Ω (cid:2) τ δ : ∇ u − f ∗ ( τ δ ) (cid:3) d x + δ Z Ω X δ : ∇ u d x +(1 − p ) δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x + δp Z Ω (cid:0) | ∂ u δ | (cid:1) p − d x . Here, as δ →
0, the second integral (recall (4.8)) and the last integral on theright-hand side converge to 0 and we obtain as δ → − f ∗ ) R [ κ ] ≤ inf u ∈C Sob J [ u ] ≤ J [ u δ ] ≤ J δ [ u δ ] → R [ σ ] + (1 − p ) δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x . Hence we have the minimizing property of the sequence { u δ } and additionally δ Z Ω (cid:0) | ∂ u δ | (cid:1) p d x → δ → . Step 2. Caccioppoli-type inequality.
Proceeding with the proof of Theorem4.3 we note that by u ∈ W ∞ (Ω)sup δ k u δ k L ∞ (Ω) < ∞ . As usual we let Γ i,δ := 1 + | ∂ i u δ | , i = 1 , . We note that for proving ii ) of Theorem 4.3, we also need an iteratedCaccioppoli-type inequality with negative exponents.23 roposition 4.1. Fix l ∈ N and suppose that η ∈ C ∞ (Ω) , ≤ η ≤ . Then,given the assumptions at the beginning of Section 4.2, the inequality Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) η l Γ α ,δ d x ≤ c Z Ω D f δ ( ∇ u δ )( ∇ η, ∇ η ) η l − Γ α +12 ,δ d x (4.10) holds for any α > − / , which in particular implies (again for all α > − / ) Z Ω η l Γ α + p − ,δ | ∂ u δ | d x ≤ c " Z Ω |∇ η | η l − Γ α +11 − γ ,δ d x + Z Ω |∇ η | η l − Γ α +1+ p − ,δ d x . (4.11) Here and in what follows c is a finite constant independent of δ ( c = c ( l, α ) ).Proof of Proposition 4.1. Let us first suppose that − / < α ≤ Z Ω Df δ ( ∇ u δ ) · ∇ ϕ d x for all ϕ ∈ C ∞ (Ω)by inserting ϕ = ∂ ψ as test function, hence0 = Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ψ (cid:1) d x for all ψ ∈ C ∞ (Ω) . With the choice ψ := η l ∂ u δ Γ α ,δ we obtain Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) η l Γ α ,δ d x = − Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ Γ α ,δ (cid:1) ∂ u δ η l d x − Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ( η l ) (cid:1) ∂ u δ Γ α ,δ d x =: S + S , (4.12)where we note | S | = 2 | α | Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) | ∂ u δ | Γ α − ,δ η l d x ≤ | α | Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) Γ α ,δ η l d x . | α | <
1, we may absorb | S | on the left-hand side of (4.12) with theresult Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) η l Γ α ,δ d x ≤ c | S | . (4.13)In the case α > Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ Γ α ,δ (cid:1) ∂ u δ η l d x ≥ . The right-hand side of (4.13) is estimated with the help of the Cauchy-Schwarz inequality which gives for any ε > Z Ω D f δ ( ∇ u δ )( ∇ ∂ u δ , ∇ η ) η l − Γ α ,δ ∂ u δ d x ≤ ε Z Ω D f δ ( ∇ u δ )( ∇ ∂ u δ , ∇ ∂ u δ ) η l Γ α ,δ d x + c ( ε ) Z Ω D f δ ( ∇ u δ )( ∇ η, ∇ η ) η l − Γ α ,δ | ∂ u δ | d x and we have (4.10) by choosing ε sufficiently small.We estimate (recall (4.2) and (4.3)) Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ η, ∇ η (cid:1) η l − Γ α +12 ,δ d x ≤ c " Z Ω |∇ η | η l − Γ γ ,δ Γ α +12 ,δ d x + Z Ω |∇ η | η l − Γ α +1+ p − ,δ d x . (4.14)Let γ > p , p via1 < p = 1 γ , p = 11 − γ . Using Young’s inequality we obtain for the first integral on the right-handside of (4.14) Z Ω |∇ η | η l − Γ γ ,δ Γ α +12 ,δ d x = Z Ω h |∇ η | η l − i p Γ γ ,δ h |∇ η | η l − i p Γ α +12 ,δ d x ≤ c " Z Ω |∇ η | η l − Γ α +11 − γ ,δ d x . (4.15)25ith (4.14) and (4.15) the proof of Proposition 4.1 is finished by observing Z Ω η l Γ α + p − ,δ | ∂ u δ | d x ≤ c Z Ω f ′′ ( ∂ u δ ) | ∂ u δ | η l Γ α ,δ d x ≤ c Z Ω " f ′′ ,δ ( ∂ u δ ) | ∂ u δ | + f ′′ ,δ ( ∂ u δ ) | ∂ u δ | η l Γ α ,δ d x ≤ c Z Ω D f δ ( ∇ ∂ u δ , ∇ ∂ u δ ) η l Γ α ,δ d x . Step 3. Main inequality.
Proposition 4.2.
Given the above hypotheses let for some τ s > , τ α > ≤ s := p −
22 + τ s , α := −
12 + τ α . Then for l sufficiently large and a local constant c ( η, l ) independent of δ , itholds Z Ω η l Γ s +12 ,δ d x ≤ c " Z Ω |∇ η | η l − Γ α +11 − γ ,δ d x + Z Ω |∇ η | η l − Γ α +1+ p − ,δ d x + Z Ω Γ s − α − p − ,δ η l d x . (4.16) Proof of Proposition 4.2.
Let us first note that on the left-hand side of (4.16)we have s + 1 = ( p/
2) + τ s . Moreover, since k u δ k L ∞ (Ω) ≤ c , we estimate Z Ω | ∂ u δ | Γ s ,δ η l d x = Z Ω ∂ u δ ∂ u δ Γ s ,δ η l d x = − Z Ω u δ ∂ h ∂ u δ Γ s ,δ η l i d x ≤ c " Z Ω | ∂ u δ | Γ s ,δ η l d x + Z Ω | ∂ u δ | η l − |∇ η | Γ s ,δ d x + Z Ω Γ s − ,δ | ∂ u δ | | ∂ u δ | η l d x ≤ c " Z Ω | ∂ u δ | Γ s ,δ η l d x + ε Z Ω | ∂ u δ | Γ s ,δ η l d x + c ( ε ) Z Ω |∇ η | η l − Γ s ,δ d x , leading to ( ε > Z Ω | ∂ u δ | Γ s ,δ η l d x ≤ c " Z Ω | ∂ u δ | Γ s ,δ η l d x + Z Ω |∇ η | η l − Γ s ,δ d x . (4.17)26he first term on the right-hand side of (4.17) is estimated with the help ofYoung’s inequality Z Ω | ∂ u δ | Γ s ,δ η l d x = Z Ω | ∂ u δ | Γ α + p − ,δ Γ s − α − p − ,δ η l d x ≤ c " Z Ω | ∂ u δ | Γ α + p − ,δ η l d x + Z Ω Γ s − α − p − ,δ η l d x . (4.18)Now (4.11) is applied to the first term on the right-hand side of (4.18) whichgives Z Ω | ∂ u δ | Γ s ,δ η l d x ≤ c " Z Ω |∇ η | η l − Γ α +11 − γ ,δ d x + Z Ω |∇ η | η l − Γ α +1+ p − ,δ d x + Z Ω Γ s − α − p − ,δ η l d x . (4.19)Combining (4.17) and (4.19) we finally obtain Z Ω Γ s +12 ,δ η l d x ≤ c " Z Ω (cid:0) η l + |∇ η | η l − (cid:1) Γ s ,δ d x + Z Ω |∇ η | η l − Γ α +11 − γ ,δ + Z Ω |∇ η | η l − Γ α +1+ p − ,δ d x + Z Ω Γ s − α − p − ,δ η l d x . (4.20)The first intergral on the right-hand side of (4.20) can be absorbed in theleft-hand side whenever l is sufficiently large (compare, e.g., [1], Proof ofProposition 2.2) which completes the proof of the proposition. Step 4. Conclusion.
To finish the proof of Theorem 4.3 we observe that wemay exactly follow the lines of [1], Theorem 1.3, provided that α + 1 + p −
22 = α + p < s + 1 , s − α − p − < s + 1 , (4.21)and provided that α + 1 < ( s + 1)(1 − γ ) = s + 1 − ( s + 1) γ . (4.22)We note that (4.21) is equivalent to τ α < τ s + 12 , τ s < τ α + 12 , i.e. | τ s − τ α | < . (4.23)27ondition (4.22) turns into −
12 + τ α < p −
22 + τ s − p + 2 τ s γ , which can be written as γ < p − τ s − τ α ) p + 2 τ s . (4.24)i) If γ = 0, then (4.23) and (4.24) are satisfied for any τ s > / τ α = τ s − /
4, which implies the first claim of the theorem.ii) If we have (4.4) with γ >
0, then we choose τ α > τ s = + τ α . Then we have (4.23) and (4.24) for τ α sufficiently smallon account of our assumption (4.4). We note that with the notationintroduced above we have χ = 2( s + 1) = 2 " p −
22 + τ s + 1 = p + 2 τ s . The choice τ s > / p -growth for arbitrary p inthe following sense: do we have higher integrability of ∂ u and simulaneouslyof ∂ u ? In fact, in addition to Theorem 4.3 we have Theorem 4.4.
Suppose that we have (1.4) and the first condition of (1.5),i.e. a linear growth condition for f . Moreover, suppose that we have (1.12)with < µ and γ = 0 and let assumption (4.3) hold.Then there exists u ∈ M such that for any κ < ∞ ∂ u ∈ L κ (Ω) . Proof of Theorem 4.4.
Since we already discussed Theorem 4.1 covering thecase p = 2 we may suppose that p >
2. Given the regularization of Theorem4.3 we need a counterpart for Propostion 4.1 with ∂ u δ rplaced by ∂ u δ andΓ ,δ by Γ ,δ . Proposition 4.3.
Given the hypotheses of Theorem 4.4 we have for any realnumbers χ > , α > − / Z Ω η l Γ α − µ ,δ | ∂ u δ | d x ≤ c " Z Ω |∇ η | η l − Γ α +11 ,δ d x + Z Ω |∇ η | η l − Γ ( α +1) χχ − ( p − ,δ d x . (4.25)28 roof of Proposition 4.3 . The counterpart of (4.14) reads as Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ η, ∇ η (cid:1) η l − Γ α +11 ,δ d x ≤ c " Z Ω |∇ η | η l − Γ α +11 ,δ d x + Z Ω |∇ η | η l − Γ α +11 ,δ Γ p − ,δ d x . (4.26)Now we let (recall p > p = χχ − ( p − , < p = χp − . For the choice of χ we have Theorem 4.3, i ), in mind. Now Young’s inequalityimplies for the second integral on the right-hand side of (4.26) Z Ω |∇ η | η l − Γ α +11 ,δ Γ p − ,δ d x = Z Ω h |∇ η | η l − i p Γ α +11 ,δ h |∇ η | η l − i p Γ p − ,δ d x ≤ c " Z Ω |∇ η | η l − Γ ( α +1) χχ − ( p − ,δ d x . (4.27)By (4.26) and (4.27) we have (4.25), i.e. the proposition is proved.Now a variant of Proposition 4.1 is needed. Proposition 4.4.
Given the hypotheses of Theorem 4.4 let for some τ s > , τ α > ≤ s := −
12 + τ s , α := −
12 + τ α . Then for l sufficiently large and a local constant c ( η, l ) independent of δ itholds Z Ω η l Γ s +11 ,δ d x ≤ c " Z Ω |∇ η | η l − Γ α +11 ,δ d x + Z Ω |∇ η | η l − Γ ( α +1) χχ − ( p − ,δ d x + Z Ω Γ s − α + µ ,δ η l d x . (4.28) Proof of Proposition 4.4.
Here we have instead of (4.17), (4.18) Z Ω | ∂ u δ | Γ s ,δ η l d x ≤ c " Z Ω | ∂ u δ | Γ s ,δ η l d x + Z Ω |∇ η | η l − Γ s ,δ d x (4.29)29nd Z Ω | ∂ u δ | Γ s ,δ η l d x = Z Ω | ∂ u δ | Γ α − µ ,δ Γ s − α + µ ,δ η l d x ≤ c " Z Ω | ∂ u δ | Γ α − µ ,δ η l d x + Z Ω Γ s − α + µ ,δ η l d x . (4.30)We then obtain Z Ω Γ s +11 ,δ η l d x ≤ c " Z Ω (cid:0) η l + |∇ η | η l − (cid:1) Γ s ,δ d x + Z Ω |∇ η | η l − Γ α +11 ,δ + Z Ω |∇ η | η l − Γ ( α +1) χχ − ( p − ,δ d x + Z Ω Γ s − α + µ ,δ η l d x , (4.31)and (4.31) gives the proposition.Finally we arrive at the following choice of parameters. α + 1 < s + 1 ⇔ τ α < τ s , (4.32)2 s − α + µ < s + 1 ⇔ τ s − τ α < − µ . (4.33)Moreover, the inequality "
12 + τ α χχ − ( p − <
12 + τ s (4.34)needs to be true.For any given τ s > τ α > µ < χ sufficientlylarge such that (4.34) is satisfied as well. This proves the theorem. Remark 4.2.
We note that even higher integrabilty of ∂ u with some χ >p + 1 (compare Theorem 4.3, ii )) would give some higher integrability of ∂ u by choosing τ α := 3(2 − µ ) − ( p − p − whenever − µ ) − ( p − > . This indicates that we also may provevariants of Theorem 4.4 with some weaker hypotheses. This is left to thereader. Corollary 4.2.
Suppose that we have the hypotheses of Theorem 4.4. Thenany weak L -cluster-point of the regularizing sequence { u δ } is of class C ,α (Ω) . The first uniqueness result is already established in Theorem 4.1 and Corol-lary 4.1.We now finally discuss the case that the superlinear part of the energy densityunder consideration is given in terms of a N -function.In this case the uniqueness of solutions to the generalized problems can beestablished without using the results of the previous section.The reason is the existence of an unique dual solution which, by the dualityrelation, can be carried over to establish the uniqueness of generalized mini-mizers.So let us suppose that we have the situation as described in the hypothesesof Theorem 3.2, in particular σ := Df (cid:0) ∇ a u (cid:1) = (cid:16) f ′ (cid:0) ∂ a u (cid:1) , A ′ (cid:0) | ∂ u | (cid:1)(cid:17) is a solution of the dual variational problem (3.11) whenever u is a general-ized minimizer in the sense of (1.11).Now, if we have a closer look at the arguments from measure theory leadingto Theorem 7 of [21], then we may adapt the proof to the situation at handand obtain: Theorem 5.1.
Suppose that we have the hypotheses stated in the beginningof Section 4.2 in front of Theorem 4.3. Moreover, suppose that u ∗ ∈ M isa L -cluster-point of a sequence { u δ } from u + ◦ W (Ω) such that for some κ ∈ R the functions h δ := h δ + h δ := (cid:0) | ∂ u δ | (cid:1) κ + (cid:0) | ∂ u δ | (cid:1) κ (5.1) are of class W , loc (Ω) uniform w.r.t. δ ∈ (0 , .Then the dual problem (3.11) is uniquely solvable.
31e emphasize that Theorem 5.1 is valid without any restriction on the ex-ponent µ in (1.12). This is just needed for Corollary 5.1 below. Main idea of the proof of Theorem 5.1.
Using the notation of [13], Section2.2, we first note that the unboundedness of U := Im( ∇ f ) w.r.t. the direction ξ causes no essential changes in the previous arguments.Given the regularization { u δ } of Step 1 in the proof of Theorem 4.3, the mainchanges concern the proof of (compare p. 22 of [13]) σ δ ( x ) → σ ( x ) , δ | ∂ u δ ( x ) | p − → , (5.2)on Σ as δ →
0. In [13], these convergences follow form the uniform local W -regularity of σ δ which in [13] is a consequence of D f ( ξ ) (cid:0) η, η ) ≤ (cid:0) | ξ | (cid:1) − | η | , ξ, η ∈ R . (5.3)Since (5.3) in general is no longer valid, we use (5.1) to get the a.e. conver-gence of ∇ u δ .We finally have Corollary 5.1.
Suppose that we have the assumptions of Theorem 4.4 with p > . Then we have the uniqueness of generalized solutions of problem(1.11).Proof of Corollary 5.1. We first establish the uniqueness of the dual solution σ . To this purpose we recall (4.11) and (4.25), i.e. given the regularizationintroduced in Step 1 of the proof of Theorem 4.3 we have choosing α = 0 Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) η d x ≤ c " Z Ω |∇ η | Γ ,δ d x + Z Ω |∇ η | Γ χχ − ( p − ,δ d x (5.4)as well as Z Ω D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) η d x ≤ c " Z Ω |∇ η | Γ ,δ d x + Z Ω |∇ η | Γ p ,δ d x . (5.5)32ow we compute Z Ω η |∇ h δ | d x ≤ c Z Ω η l (cid:0) | ∂ u | (cid:1) κ − |∇ ∂ u δ | d x + c Z Ω η (cid:0) | ∂ u | (cid:1) κ − |∇ ∂ u δ | d x ≤ c Z Ω η h Γ − µ ,δ | ∂ ∂ u δ | + | ∂ ∂ u δ | i Γ µ + κ − ,δ d x + c Z Ω η h Γ − µ ,δ | ∂ ∂ u δ | + | ∂ ∂ u δ | i Γ µ ,δ Γ κ − ,δ d x which leads to Z Ω η |∇ h δ | d x ≤ Z Ω η D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) d x + Z Ω η D f δ ( ∇ u δ ) (cid:0) ∇ ∂ u δ , ∇ ∂ u δ (cid:1) d x + Z Ω η Γ t ,δ d x + Z Ω η Γ t ,δ d x (5.6)with some finite exponents t and t . By (5.4) - (5.6) and Theorem 4.3, i ),Theorem 4.4 we have that h δ is uniformly of class W , loc which by Theorem5.1 implies the uniqueness of the dual solution.The next step for proving Corollary 5.1 is to use Corollary 4.2 to obtain C ,α -regularity of the dual solution. Then, as outlined in the proof of TheoremA.9 in [13], a suitable comparison argument w.r.t. σ can be carried out toobtain for any generalized minimizer u ∈ C BV ∇ u = ∇ f ∗ ( σ ) . (5.7)We note that (5.4) in particular gives ∇ s u = 0 . By the uniqueness of σ the uniqueness of generalized minimizers up to anadditive constant is established. Finally, on account ot u ∈ C BV we have( u − u ) ν = 0 H a.e. on ∂ Ω and in conclusion the uniqueness of generalizedminimizers.
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