bilđại họcauer m convex variational problems sinhvienzone com

1818 Lecture Notes in Mathematics Si nh Vi en Zo ne C om Editors: J. M Morel, Cachan F Takens, Groningen B Teissier, Paris SinhVienZone.com https://fb.com/sinhvienzonevn .C om ne Zo en Vi nh Si Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo SinhVienZone.com https://fb.com/sinhvienzonevn Michael Bildhauer Convex Variational Problems Si nh Vi en Zo ne C om Linear, Nearly Linear and Anisotropic Growth Conditions 13 SinhVienZone.com https://fb.com/sinhvienzonevn Author C om Michael Bildhauer Department of Mathematics Saarland University P.O Box 151150 66041 Saarbrăucken Germany e-mail: bibi@math.uni-sb.de Cataloging-in-Publication Data applied for Vi en Zo ne Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 49-02, 49N60, 49N15, 35-02, 35J20, 35J50 Si nh ISSN 0075-8434 ISBN 3-540-40298-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de 2c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready TEX output by the authors SPIN: 10935590 SinhVienZone.com 41/3142/du - 543210 - Printed on acid-free paper https://fb.com/sinhvienzonevn Preface C om In recent years, two (at first glance) quite different fields of mathematical interest have attracted my attention ne • Elliptic variational problems with linear growth conditions Here the notion of a “solution” is not obvious and, in fact, the point of view has to be changed several times in order to get some deeper insight • The study of the smoothness properties of solutions to convex anisotropic variational problems with superlinear growth Vi en Zo It took some time to realize that, in spite of the fundamental differences and with the help of some suitable theorems on the existence and uniqueness of solutions in the case of linear growth conditions, a non-uniform ellipticity condition serves as the main tool towards a unified view of the regularity theory for both kinds of problems This is roughly speaking the background of my habilitations thesis at the Saarland University which is the basis for this presentation Si nh Of course there is a long list of people who have contributed to this monograph in one or the other way and I express my thanks to each of them Without trying to list them all, I really want to mention: Prof G Mingione is one of the authors of the joint paper [BFM] The valuable discussions on variational problems with non-standard growth conditions go much beyond this publication Prof G Seregin took this part in the case of variational problems with linear growth Large parts of the presented material are joint work with Prof M Fuchs: this, in the best possible sense, requires no further comment Moreover, I am deeply grateful for the numerous discussions and the helpful suggestions Saarbră ucken, April 2003 SinhVienZone.com Michael Bildhauer https://fb.com/sinhvienzonevn .C om ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn Si nh Vi en Zo ne C om Dedicated to Christina SinhVienZone.com https://fb.com/sinhvienzonevn .C om ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn Contents Introduction Variational problems with linear growth: the general setting 2.1 Construction of a solution for the dual problem which is of (Ω; RnN ) class W2,loc 2.1.1 The dual problem 2.1.2 Regularization -regularity for the dual problem 2.1.3 W2,loc 2.2 A uniqueness theorem for the dual problem 2.3 Partial C 1,2 - and C 0,2 -regularity, respectively, for generalized minimizers and for the dual solution 2.3.1 Partial C 1,2 -regularity of generalized minimizers 2.3.2 Partial C 0,2 -regularity of the dual solution 2.4 Degenerate variational problems with linear growth 2.4.1 The duality relation for degenerate problems 2.4.2 Application: an intrinsic regularity theory for 13 nh Variational integrands with (s, μ, q)-growth 3.1 Existence in Orlicz-Sobolev spaces 3.2 The notion of (s, μ, q)-growth – examples 3.3 A priori gradient bounds and local C 1,2 -estimates for scalar and structured vector-valued problems 3.3.1 Regularization 3.3.2 A priori Lq -estimates 3.3.3 Proof of Theorem 3.16 3.3.4 Conclusion 3.4 Partial regularity in the general vectorial setting 3.4.1 Regularization 3.4.2 A Caccioppoli-type inequality 3.4.3 Blow-up 3.4.3.1 Blow-up and limit equation 3.4.3.2 An auxiliary proposition 3.4.3.3 Strong convergence 3.4.3.4 Conclusion 3.4.4 Iteration Si Vi en Zo ne C om SinhVienZone.com https://fb.com/sinhvienzonevn 14 14 16 19 20 25 26 29 32 33 39 41 42 44 50 52 54 61 67 69 69 70 72 74 76 83 86 87 X Contents 3.5 Comparison with some known results 3.5.1 The scalar case 3.5.2 The vectorial setting 3.6 Two-dimensional anisotropic variational problems 89 89 90 91 Variational problems with linear growth: the case of μ-elliptic integrands 97 4.1 The case μ < + 2/n 100 4.1.1 Regularization 101 4.1.2 Some remarks on the dual problem 101 4.1.3 Proof of Theorem 4.4 103 4.2 Bounded generalized solutions 104 4.2.1 Regularization 108 4.2.2 The limit case μ = 111 4.2.2.1 Higher local integrability 111 4.2.2.2 The independent variable 113 4.2.3 Lp -estimates in the case μ < 116 4.2.4 A priori gradient bounds 118 4.3 Two-dimensional problems 122 4.3.1 Higher local integrability in the limit case 123 4.3.2 The case μ < 129 4.4 A counterexample 132 Bounded solutions for convex variational problems with a wide range of anisotropy 141 5.1 Vector-valued problems 142 5.2 Scalar obstacle problems 149 Anisotropic linear/superlinear growth in the scalar case 161 A Some remarks on relaxation 173 A.1 The approach known from the minimal surface case 174 A.2 The approach known from the theory of perfect plasticity 176 A.3 Two uniqueness results 181 B Si nh Vi en Zo ne C om Some density results 185 B.1 Approximations in BV 185 B.2 A density result for U L(c) 191 B.3 Local comparison functions 194 C Brief comments on steady states of generalized Newtonian fluids 199 D Notation and conventions 205 References 207 Index 215 SinhVienZone.com https://fb.com/sinhvienzonevn 2 − g(6 w) := sup w div dx − g (2 ) dx (3) C02 (U ;RnN ), |2 |≤1 U U U Now we follow [Giu2], pp 14, fix ε > and w BV (Ω; RN ) Moreover, for any l N we let   l k = k := x Ω : dist (x, ∂Ω) > , k = 0, 1, 2, l+k −2 C om Here l may be chosen sufficiently large such that |6 w| < (4) Given k as above, sets Ai are defined by induction: A1 := and Ai := i+1 − i−1 , i = 2, 3, Zo ne With respect to these Ai we then consider a partition of the unity {ϕi }, i.e for any i N + ϕi C0 (Ai ) , ≤ ϕi ≤ , ϕi = i=1 nh Vi en Finally, η denotes a smoothing kernel and we choose for any i N a positive number i sufficiently small such that (letting −1 = ∅) 2 spt ηεi ∗ (ϕi w) 3 i+2 − i−2 ;   ηε ∗ (ϕi w) − ϕi w dx < 2−i ; i (5)   ηε ∗ (w ⊗ ϕi ) − w ⊗ ϕi  dx < 2−i ε i Si 2 Now assume that = m and let wm = + ηεi ∗ (ϕi w) i=1 Note that here and in the following each sum under consideration is locally finite The first claim of the lemma is immediate since {wm } is constructed as a smooth sequence and since we have recalling (5) 2 +  ηε ∗ (ϕi w) − ϕi w| dx ≤ ε |wm − w| dx ≤ i SinhVienZone.com i=1 https://fb.com/sinhvienzonevn B.1 Approximations in BV This, together with lower semicontinuity, also proves 2 g(6 w) ≤ lim inf g(6 wm ) dx m→5 187 (6) To establish the opposite inequality we fix C05 (Ω; RnN ), |2 | ≤ Then, 15 by (4), (5), on account of i=1 ϕi ≡ and since the intersection of more than three Ai is empty, we obtain 2 wm div dx − g (2 ) dx − 2 2 + ⎬ ⎬ ⎫ ⎫ = − w div ϕ1 ηε1 ∗ dx − w div ϕi ηεi ∗ dx i=2 ⎬ ⎫ : ηεi ∗ (w ⊗ ϕi ) − w ⊗ ϕi dx − i=1 2 ⎫ ⎬ w div ϕ1 ηε1 ∗ dx − g (2 ) dx + ε ≤ − (7) C om + +2 g (2 ) dx Zo ne If we identify with its zero-extension to Rn , then we may write 2 − g (2 ) dx = − g (2 ) dx n R  2   =− g (ηε1 ∗ ) − g (2 ) dx g (ηε1 ∗ ) dx + Rn en Rn = I + II (8) nh Vi An estimate for I follows from ≤ ϕ1 ≤ 1, the convexity of g , g (0) = and g 0: 2 I ≤ − g (ϕ1 ηε1 ∗ ) dx = − g (ϕ1 ηε1 ∗ ) dx (9) Si Rn To handle the second integral of (8) we use Jensen’s inequality g (ηε1 ∗ ) ≤ ηε1 ∗ g (2 ) This, together with standard calculations, gives   ηε1 ∗ g (2 ) − g (2 ) dx = II ≤ (10) Rn As a result of (7)–(10) it is proved that 2 wm div dx − g (2 ) dx − 2 2 ⎫ ⎬ w div ϕ1 ηε1 ∗ dx − g (ϕ1 ηε1 ∗ ) dx + ε ≤ − SinhVienZone.com https://fb.com/sinhvienzonevn (11) 188 B Some density results Now it is obvious that the comparison function on the right-hand side of (11) is admissible and the opposite inequality to (6) is established: 2 g(6 wm ) dx ≤ g(6 w) lim sup m→5 2 Moreover, the approximative functions wm are defined in the same manner as in [Giu2], pp 14, hence Remark 2.12 of [Giu2], p 38, remains valid which  proves that the trace of each wm on ∂3 coincides with the trace of w  As a generalization of Lemma B.1 we now choose an approximative sequence with arbitrary prescribed traces To this purpose we fix again a bounded Lipschitz domain 3ˆ and extend our boundary values u0 ˆ RN ) W (Ω; RN ) to a function of class W (Ω; C om ˆ Lemma B.2 Given 3ˆ , u0 as above let w BV (Ω; RN ) and denote by w ˆ the extension via u0 to the domain Then there exists a sequence {wm } in u0 |2 + C05 (Ω; RN ) such that if m → ∞ (and if we extend wm by u0 to 3ˆ ) ii) ˆ RN ); ˆ in L1 (Ω; wm w 2 + |6 wm |2 dx + |6 w| ˆ 2ˆ Zo 2ˆ ne i) Vi en Proof Let us recall the general setting (1)–(3) and fix w, w ˆ as given in the lemma Now we use a construction as outlined in [Alt], p 170 Since ∂Ω is Lipschitz, we can cover ∂3 by open sets V1 , , Vr such that after rotation and translation Vj takes the form       n     Vj = x R : (x1 , , xn−1 ) < rj , xn − gj (x1 , , xn−1 ) < hj , nh where gj is a Lipschitz function Moreover, we have ⇒ x ∂Ω ; < xn − gj (x1 , , xn−1 ) < hj ⇒x7 Ω; Si = xn − gj (x1 , , xn−1 ) > xn − gj (x1 , , xn−1 ) > −hj ⇒ x Let Vr+1 , Vr+2 denote open sets such that V r+1 Ω, V r+2 Rn − Ω and 3ˆ r+2 / Vj j=1 In addition, the sets Vj may be chosen such that for any j = 1, , r + g(6 w)(∂V ˆ j) = SinhVienZone.com https://fb.com/sinhvienzonevn B.1 Approximations in BV 189 Here we let for any Borel set B 3ˆ (again following [DT], see also the proof of Theorem A.6) g(6 w) ˆ g(6 w)(B) ˆ = B = sup ˆ nN ), |2 |≤1 C02 (Ω;R 2ˆ 1B w ˆ− g (2 ) dx 2ˆ r+2 + C om Now fix ε > and decrease Vj if necessary (to be a little bit more precise: decrease radius and height of Vj a little bit and obtain sets Vj,ε Vj Then let U1 = V1 , U2 = V2 − V 1,ε and inductively define Uj , j = 1, , r + 2.) As a result we obtain smooth sets Uj which can be arranged in addition such that ˆ +ε, g(6 w)(U ˆ j ) ≤ g(6 w)( ˆ Ω) j=1 g(6 w)(∂U ˆ j) = , j = 1, , r + (12) (13) ≤ ϕj ≤ , (Uj ) , r+2 + ϕj ≡ on 3ˆ j=0 en ϕj C05 Zo ne ˆ RN ), hence we may extend w ˆ Suborˆ by to Rn − Ω Note that u0 W11 (Ω; dinate to the covering {Uj } let {ϕj } denote a partition of the unity, i.e nh Vi For each fixed index ≤ j ≤ r and for δ we then define (neglecting with a slight abuse of notation rotations and translations)   vˆjδ,ε (x) := ϕj (x) (w ˆ − u0 ) ◦ (x − δ en ) + u0 (x) Si δ,ε δ,ε Finally, we let vˆr+1 := ϕr+1 w, vˆr+2 := ϕr+2 u0 and δ,ε vˆ := r+2 + vˆjδ,ε j=1 Clearly vˆδ,ε − u0 is compactly supported in Moreover, for δ = δ(6 ) su5ciently small we have   (ε),ε vˆ −w ˆ  dx ≤ ε , 2ˆ and again lower semicontinuity gives for vˆε = vˆ5 (ε),ε 2 g(6 w) ˆ ≤ lim inf g(6 vˆε ) dx 2ˆ ε→0 (14) 2ˆ ˆ RnN ), To prove the opposite inequality we recall the definition (3), fix C05 (Ω; |2 | ≤ 1, and observe SinhVienZone.com https://fb.com/sinhvienzonevn B Some density results − vˆ div dx = − ε 2ˆ r + j=1 − Uj r+2 + j=r+1 + r + j=1 ≤− ϕj (x) w(x) ˆ div dx Uj   ⎫ ⎬ dx : ϕj u0 (x) − u0 x − δ(6 ) en Uj r + j=1 − ⎬ ⎫ ϕj (x) w ˆ x − δ(6 ) en div dx ⎬ ⎫ ϕj (x) w ˆ x − δ(6 ) en div dx Uj r+2 + j=r+1 ϕj (x) w(x) ˆ div dx + ε , C om 190 Uj where δ(6 ) is assumed to be su5ciently small Note that this choice does not depend on Now, the right-hand side is estimated by + ⎬ ⎫ w ˆ x − δ(6 ) en ⊗ ϕj : dx Uj r+2 + Uj j=r+1 w(x) ˆ ⊗ ϕj : dx + Uj Uj Si r+2 + j=r+1 + r+2 + ⎬ ⎫ w ˆ x − δ(6 ) en div (ϕj ) dx nh r + j=1 − w(x) ˆ div (ϕj ) dx + Vi j=r+1 ≤− ne r + j=1 − Uj Zo j=1 ⎬ ⎫ w ˆ x − δ(6 ) en div (ϕj ) dx en r.h.s = − r + r+2 + j=1 Uj j=1 Uj w(x) ˆ div (ϕj ) dx Uj w(x) ˆ ⊗ ϕj : dx r +  ⎫ ⎬ w(x) ˆ −w ˆ x − δ(6 ) en  |6 ϕj | dx + ε + Once more δ(6 ) is decreased in order to bound the last sum by Moreover, 1r+2 j=1 ϕj ≡ implies the third sum to vanish identically Next observe that g (ϕj ) ≤ ϕj g (2 ) follows from the convexity of g ˆ and from g (0) = 0, thus we arrive at (2 ≡ on Rn − Ω) SinhVienZone.com https://fb.com/sinhvienzonevn B.2 A density result for U L(c) 2 vˆ div dx − g (2 ) dx 2ˆ 2ˆ   2 r + ⎬ ⎫ − ≤ w ˆ x − δ(6 ) en div (ϕj ) dx − g (ϕj ) dx ε Uj j=1 + (15) Uj   − w(x) ˆ div (ϕj ) dx − g (ϕj ) dx + ε r+2 + Uj i=r+1 Uj ˜j Finally, we choose open sets U r + Uj , j = 1, , r such that r + ˜j ) ≤ g(6 w)( ˆ U j=1 g(6 w)(U ˆ j) + ε (16) j=1 C om − 191 Note that sets of this kind can be found on account of (13) Moreover, if δ(6 ) 1, then we remark that for j = 1, , r sup ˜j ;RnN ), |2˜ |≤1 2˜ C02 (U − g (ϕj ) dx Uj ˜j U w ˆ div 2˜ dx − Zo ≤ ne ⎬ ⎫ − w ˆ x − δ(6 ) en div (ϕj ) dx − Uj  ˜j U  g (2˜ ) dx (17) vˆ div dx − 2ˆ nh ≤ ≤ Si ε r + Vi − en Putting together the inequalities (15)–(17) it is proved that g (2 ) dx 2ˆ ˜j ) + g(6 w)( ˆ U j=1 r+2 + r+2 + g(6 w)(U ˆ j) + 26 (18) j=r+1 g(6 w)(U ˆ j) + 3ε j=r+1 Once (18) is established, (12) shows the opposite inequality to (14) Summarizing the results we have proved up to now: there exists a sequence {ˆ vε } such that vˆε − u0 is compactly supported in and such that the convergences claimed in the lemma hold for this sequence In a last step it remains to apply the standard smoothing procedure (see Lemma B.1) and Lemma B.2 is proved   B.2 A density result for U L(c) Here we are going to establish a density result from [BF4] which was needed for the proof of the identification Theorem A.6 With the notation U and L(c) as introduced in Appendix A.2, a precise formulation of this lemma reads as SinhVienZone.com https://fb.com/sinhvienzonevn 192 B Some density results Lemma B.3 Suppose that ∈ U satisfies (x) L(c) for some c R Then a sequence {2 m }, m C (Ω; RnN ) exists such that ⎫ ⎬ i) m in Lt Ω; RnN for any t < ∞ and we have almost everywhere convergence; ii) div ⎫ ⎬ div in Ln Ω; RN ; m  in L5 iii) m iv) m (x) ⎫ nN ⎬ Ω; R ; L(c) for all x and for any m N 3 r / C om Proof As in the proof of Lemma B.2, the boundary of the Lipschitz domain is covered with sets Vj , j = 1, , r, such that we have the properties stated there Let V0 denote an open set satisfying V Ω, Vj , j=0 (Vj ) , ≤ ϕj ≤ , Zo ϕj C05 ne and consider a corresponding partition of the unity {ϕj }, i.e r + ϕj ≡ on j=0 nh Vi en For each fixed index j we let for δ ⎨ (x + δ e ) ϕ (x) if x V ; n j j j (x) := ⎩ if x − Vj Si Note that j5 ≡ near the “upper” boundary part of Vj , the same is true near the “vertical” boundary parts which follows from the support properties of ϕj and from an appropriate choice of δ If ω denotes a smoothing kernel, we let r   + δ,ρ j (x) , x (x) := ωρ ∗ ϕ0 + j=1 Assuming again the standard representation of the neighborhood Vj we get ωρ (y − x) (y + δ en ) ϕj (y) dy , ωρ ∗ j (x) = Rn and for ρ small enough depending on δ we see that for y Bρ (x), x Ω, the point y + δ en belongs to 3, and ωρ ∗ j5 (x) is well defined Clearly ωρ ∗ j5 C (Ω; RnN ) and ⎫ ⎬ ρ↓0 ωρ ∗ j5 j5 in Lp Ω; RnN SinhVienZone.com https://fb.com/sinhvienzonevn B.2 A density result for U L(c) 193 for any p < ∞, moreover (see [Alt], Lemma 1.16, p 18) 5 ↓0 j ⎫ ⎬ in Lp Ω; RnN , ϕj again for any p < ∞ We further have for x ⎬ ⎫ div ωρ ∗ j5 (x) = − ⎫ ⎬ ∂2 ωρ (y − x) 2 (y + δ en ) ϕj (y) dy B2 (x) and since it is sufficient to consider x Vj , we see that the arguments on the right-hand side are compactly supported in Moreover, y 5 ωρ (y − x) has compact support in Bρ (x), thus  ωρ (y − x) div (y + δ en ) ϕj (y) C om ⎫ ⎬ div ωρ ∗ j5 (x) = B2 (x)  +2 (y + δ en ) ϕj (y) dy and, as above, ne ⎫ ρ↓0 div ωρ ∗ j5 ) div (· + δ en ) ϕj + (· + δ en ) ϕj Zo in Ln (Ω; RN ) The right-hand side converges to en div ϕj + ϕj Si nh Vi in Ln (Ω; RN ) as δ ↓ So, if we first fix a sequence δm ↓ 0, we find a sequence {ρm } depending on {δm } such that the convergence properties i) and ii) hold for m := m ,ρm The boundedness of 2 m L2 (Ω;RnN ) implies m  2˜ in L5 (Ω; RnN ) for a subsequence and some tensor 2˜ L5 (Ω; RnN ), but i) shows 2˜ = It remains to prove iv) Jensen’s inequality applied to the measure ωρ (x − ·)Ln gives ⎫ ⎬ f δ,ρ (x) ≤ ωρ (x − y) f  ϕ0 + B2 (x) and if we recall the definition of combination r + j  (y) dy j=1 j we see that f is evaluated on the convex ϕ0 (y) (y) + r + ϕj (y) ( ) , j=1 where ( ) has an obvious meaning for j = 1, , r Our assumption L(c) almost everywhere then implies f  ϕ0 + r + j  (y) ≤ c , j=1 SinhVienZone.com https://fb.com/sinhvienzonevn 194 B Some density results i.e δ,ρ L(c)   For technical reasons (see the proof of Theorem A.6), we also need the following Remark B.4 Recall the definition of j5 (x) given in the proof of Lemma B.3 Clearly this definition makes sense for points x such that x + δ en , i.e −δ ≤ xn − gj (x1 , , xn−1 ) , j n j C om so that (combined with the smoothing procedure outlined above) j5 C (Vj [−δ/2 ≤ xn − gj (x1 , , xn−1 )]) If we then let ⎨ if x V , x − g (x , , x j n j n−1 ) , ψj5 (x) := ⎩ if x V , x − g (x , , x ) ≤ −δ/4 , n−1 Zo ne ψj5 C05 (Rn ), ≤ ψ ≤ 1, then the function ψj5 j5 , j = 1, , r, is of class ˆ RnN ), where 3ˆ is some bounded Lipschitz domain and δ is chosen C05 (Ω; sufficiently small As a result, if 3ˆ is fixed as above, then the sequence {2 m } ˆ RnN ) given in Lemma B.3 may be in addition assumed to be of class C05 (Ω; Moreover, again by the convexity of f (further recall that f (0) = 0), the level set property continues to hold on the extended domain 3ˆ en B.3 Local comparison functions nh Vi A helpful tool which was used in Sections 2.3, 2.4 and 4.3 is the construction of local comparison functions as given in [BF1] With the notation ˆ f (6 w) dx for any open set 3ˆ J[w; Ω] = 2ˆ Si we now prove for f given as in Assumption 2.1 Lemma B.5 Consider a sequence {um } W11 (Ω; RN ) such that: i) um u6 in L1 (Ω; RN ) as m → ∞; ii) sup um W11 (Ω;RN ) < ∞ m4 N Then we can find a subsequence (still denoted by {um }) with the following properties: for any x0 and for almost any ball BR (x0 ), B2R (x0 ) , there is a sequence {wm } W11 (Ω; RN ) such that i) wm u6 in L1 (Ω; RN ) as m → ∞;     ii) lim sup J wm ; BR (x0 ) ≤ lim inf J um ; BR (x0 ) ; m→5 SinhVienZone.com m→5 https://fb.com/sinhvienzonevn B.3 Local comparison functions 195 iii) lim sup J[wm ; 3] ≤ lim inf J[um ; 3] ; m→5 m→5 iv) wm |∂BR (x0 ) = u6|∂BR (x0 ) , where the traces are well defined functions of class L1 (∂BR (x0 ); RN ); v) wm |2 −B2R (x0 ) = umk |2 −B2R (x0 ) for any m N, in particular the boundary values on ∂3 are preserved; vi) wm |BR/2 (x0 ) = uml |BR/2 (x0 ) for any m N Here {umk } and {uml } denote some appropriate subsequences of {um } Proof We have u6 BV and we may also assume that C om u m  u6 , |6 um |  μ as m → ∞ in the sense of measures where μ denotes a Radon measure of finite mass We may choose a radius R > according to Ω and μ(∂BR (x0 )) = = |6 u6 |(∂BR (x0 )) ne B2R (x0 ) (19) Vi en Zo This implies (see [Giu2], Remark 2.13) that u6|∂BR (x0 ) is well defined With Tε := {x : R − ε < |x − x0 | < R + }, ε > sufficiently small, we further obtain: (20) lim lim sup |6 um | dx = , ε→0 m→5 Tε lim |6 u6 | = (21) Tε nh ε→0 Si For (21) we just observe using (19)   2 ε→0 |6 u6 | −5 |6 u6 | T5 = |6 u6 |(∂BR (x0 )) = Tε δ>0 Next, let ϕε C0 (T2ε , [0, 1]), ϕε = on Tε Then 2 Tε hence |6 um | dx ≤ m→5 ϕε |6 um | dx −5 T2ε ϕε dμ , T2ε ε→0 |6 um | dx ≤ μ(T2ε ) −5 lim sup m→5 μ(∂BR (x0 )) = , Tε thus (20) holds With R fixed we now let   ⎫ ⎬ N K := w W1 BR (x0 ); R : w|∂BR (x0 ) = u|BR (x0 ) SinhVienZone.com https://fb.com/sinhvienzonevn 196 B Some density results Note that K = ∅ on account of [Giu2], Remark 2.12 and [Giu2], Theorem 2.16 We first claim that there exists a sequence {vk } K for which conclusion (ii.) holds:     lim sup J vk ; BR (x0 ) ≤ lim inf J um ; BR (x0 ) (22) m→5 k→5 For proving (22) consider u ˜m C (BR (x0 ), RN ) with (compare again [Giu2], Remark 2.12) 2 |˜ um − u | dx , |6 u ˜m | dx |6 u6 | BR (x0 ) BR (x0 ) BR (x0 ) as m → ∞ and such that u ˜m|∂BR (x0 ) = u6|∂BR (x0 ) Then  |6 u ˜m | dx lim sup lim ε→0 m→5 C om  BR (x0 )∩Tε = (23) In fact, let Aε := {x : R − ε < |x − x0 | < R} We have (see [Giu2], Prop 1.13) 2 m→5 |6 u ˜m | dx = lim sup m→5 ≤ Zo BR (x0 )∩Tε |6 u ˜m | dx |6 u | ≤ |6 u6 | ne lim sup BR (x0 )∩Aε BR (x0 )∩Aε T2ε nh Vi en as and (23) follows We define ⎨ on BR (x0 ) − Tε , um ε vm := ⎩ u + −1 (˜ um − um )(|x| − R + ) on Aε = BR (x0 ) Tε m Si ε ε Both parts of vm induce the same trace on ∂BR−ε (x0 ), thus vm is of class N ε W1 (BR (x0 );R ) and in addition vm K Since f is of linear growth, the ε ε ) dx follows from the discussion of Aε |6 vm | dx: behavior of Aε f (6 vm |6 Aε ε vm | 2 dx ≤ |6 um | dx + |6 u ˜m | dx + Aε Aε |um − u ˜m | dx BR (x0 ) According to (20) and (23) it is possible to define a sequence k such that 2 1 and lim sup |6 um | dx ≤ |6 u ˜m | dx ≤ lim sup k k m→5 m→5 Aε Aε k k for all k N, thus, for any k N, there is mk N such that 2 2 and |6 um | dx ≤ |6 u ˜m | dx ≤ k k Aε Aε k SinhVienZone.com k https://fb.com/sinhvienzonevn B.3 Local comparison functions for all m mk Recalling the L1 –convergences um , u ˜m assume in addition |um − u ˜m | dx ≤ k k BR (x0 ) 197 u6 on BR (x0 ), we for all m mk , k N Putting together these estimates we get  ε  k f (6 um ) dx + k , m mk , k N , J vm ; BR (x0 ) ≤ BR (x0 )−Tεk for a sequence can arrange k 0, k as k → ∞ By enlarging mk , if necessary, we lim k→5 BR (x0 ) C om f (6 umk ) dx = lim inf l→5 εk Finally vk := vm is introduced Then k   lim sup J vk ; BR (x0 ) ≤ lim sup k→5 BR (x0 ) ne k→5 f (6 ul ) dx BR (x0 ) f (6 umk ) dx , (24) BR (x0 ) en k→5 Zo and (22) is established From the definition of vk it also follows that lim |vk − u6 | dx = nh Vi Note that by construction we clearly may assume that vk |BR/2 (x0 ) = umi |BR/2 (x0 ) for any k N and for some subsequence of {um } Next, an analogous construction is needed in the exterior domain: choose some radius ρ > R, Bρ (x0 ) Ω, and a sequence u ˆm C (Bρ (x0 ) − BR (x0 ); RN ) satisfying Si u ˆm|6 (B2 (x0 )−BR (x0 )) = u6|6 (B2 (x0 )−BR (x0 )) , and B2 (x0 )−BR (x0 ) u ˆm u6 in L1 (Bρ (x0 ) − BR (x0 ); RN ) |6 u ˆm | dx |6 u6 | as m → ∞ B2 (x0 )−BR (x0 ) For small enough ε > and for {vm }m4 N given in (22) we then let ⎪ ⎪ ⎪ vm on BR (x0 ) ⎪ ⎪ ⎪ ⎨ ε := u wm ˆm )(|x| − R) on Tε − BR (x0 ) ˆm + −1 (um − u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ on the rest of um SinhVienZone.com https://fb.com/sinhvienzonevn 198 B Some density results ε Again wm is seen to be of class W11 (Ω; RN ) and we have ε ; 3] J[wm   ε = J vm ; BR (x0 ) + f (6 wm ) dx Tε −BR (x0 ) + f (6 um ) dx ∩[|x−x0 |>R+ε]   = J vm ; BR (x0 ) + f (6 um ) dx −BR (x0 ) 2 ε + f (6 wm ) dx − f (6 um ) dx Tε −BR (x0 ) (25) Tε −BR (x0 ) C om As before we deduce from (20) the existence of sequences k and mk → ∞ such that 2 2 and |6 um | dx ≤ |6 u ˆm | dx ≤ k k Tε −BR (x0 ) Tε −BR (x0 ) k k ne for all m mk , for all k N and we get 2 6 εk |6 wm | dx ≤ + |um − u ˆm | dx ≤ + k , k k Tε −BR (x0 ) k Tε −BR (x0 ) k Zo k   ≤ J vm ; BR (x0 ) + Vi εk ; 3] J[wm en again for all k N and for all m mk Thus decomposition (25) gives together with (22) for mk chosen sufficiently large 2 −BR (x0 ) nh   ≤ lim inf J um ; BR (x0 ) + m→5 f (6 um ) dx + βk 2 −BR (x0 ) f (6 um ) dx + 2βk Si for any k N, for all m mk and with a sequence βk as k → ∞ Again enlarging mk if necessary we may assume that     εk J[wm ; 3] ≤ lim inf J u ; B (x ) + lim inf J u ; − B (x ) + 3βk m R m R k m→5 m→5 ≤ lim inf J[um ; 3] + βk m→5 εk Setting wk = wm the lemma is proved observing that as in (24) the definition k of wk also implies L1 convergence on the whole domain Moreover, it is clear that we may assume v)   SinhVienZone.com https://fb.com/sinhvienzonevn ... (Rm ) = 2m? ??1 m m−1 2m g (s) ds g   m? ??1 2m m−1 2m   m? ??1 −ε, 2m m−1 +ε 2m Combining these inequalities it is proved that f (Rm ) ≤ SinhVienZone. com f (P ) + f (Qm ) − ε 2 https://fb .com/ sinhvienzonevn... 2003 SinhVienZone. com Michael Bildhauer https://fb .com/ sinhvienzonevn .C om ne Zo en Vi nh Si SinhVienZone. com https://fb .com/ sinhvienzonevn Si nh Vi en Zo ne C om Dedicated to Christina SinhVienZone. com. .. 215 SinhVienZone. com https://fb .com/ sinhvienzonevn C om Introduction Vi en Zo ne One of the most fundamental problems arising in the calculus of variations is to minimize strictly convex energy