Holder Continuity of Local Minimizers. Giovanni Cupini, Nicola Fusco, and Raffaella Petti

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1 Journal of Mathematical Analysis and Alications 35, Article ID jmaa available online at htt:wwwidealibrarycom on older Continuity of Local Minimizers Giovanni Cuini, icola Fusco, and affaella Petti Diartimento di Matematica, Ulisse Dini, Viale Morgagni 67 A, 5134 Florence, Italy Submitted by Arrigo Cellina eceived March 19, ITODUCTIO In recent years many results have aeared concerning the regularity of minimizers of integral functionals of the tye Ž F ; F x, x, D x, 11 where F: is an integrand satisfying the growth assumtion Ž z FŽ x, u, z L z Ž 1 with L, 1 oughly seaking two kinds of results are available If no other assumtion is made on the integrand F, it is known Žsee 7 that condition 1 ensures that a W minimizer u is older continuous for some exonent deending on L, and On the other hand, if F is assumed to be smooth enough, for instance C with resect to z, and satisfies a standard elliticity assumtion of the form Ý ij i j i, j1 Ž D FŽ x, u, z z, Ž 13 one gets that Du is older continuous Žsee, eg, 3, 8, 1 If one is interested only into Lischitz continuity roerties of minimizers, the situation is somewhat different In fact a classical result due to -47X99 $3 Coyright 1999 by Academic Press All rights of reroduction in any form reserved 578

2 OLDE COTIUITY OF LOCAL MIIMIZES 579 artman and Stamacchia Žsee 11 says that at least when the integrand deends only on Du, the convexity of F, together with the so called bounded sloe condition, yields the global boundedness of the gradient of a minimizer u In the same sirit in 5 it has been roven that if F FŽ z satisfies Ž 1 and F z z f z, where and f z is a convex function such that f z L z then every local minimizer is locally Lischitz At this oint it is natural to investigate whether such a result also holds in the general case Ž 11 It is clear that now a continuity assumtion with resect to x and u should be required In fact it is well known that even in two dimensions, taking FŽ x, z ax z with ax, if ax is not continuous, then local minimizers are only -older continuous with ' Žsee 13 In this aer we study functionals of the tye Ž 11, where F is uniformly continuous in Ž x, u with resect to z Žsee condition Ž F in Section 3 3 We do not make any differentiability assumtion on F and in articular we do not require an elliticity condition of the tye Ž 13 Instead we shall assume that FŽ x, u, z can be slit as above Žuniformly with resect to Ž x, u Under these assumtions we cannot exect minimizers to be Lischitz continuous Ž see Examle 3 owever, we rove Žsee Theorem 31 that every minimizer of functional Ž 11 is locally older continuous for any 1 The roof of our result goes as follows We consider first the case when, F only deends on x and z In this case we rove that u C Ž loc for all 1 and we show that the older estimates on u only deend on the constants L and above Ž see Theorem 5 We notice that when F FŽ x, u, z we cannot reduce to the revious case by the standard device of freezing the functional with resect to the variable u, since we lack the elliticity assumtion on F needed in order to make this argument work This difficulty is instead overcome by an aroximation argument based on a variational rincile due to Ekeland PELIMIAY ESULTS In the sequel will denote a bounded oen set in, Ž x the ball x : x x 4; we shall write in lace of Ž x if no confu-

3 58 CUPII, FUSCO, AD PETTI sion may arise If f is an integrable function we set 1 f x, fž x fž x x x where is the Lebesgue measure of the ball The letter c will stand for a generic constant that may vary from line to line If u is a older continuous function on A with exonent 1 we shall denote by u the older constant of u in A, ie,, A ½ 5 už x už y u, A su : x, y A, x y x y We recall the following definition DEFIITIO 1 Let us consider the functional Ž 11 A function u W Ž is a Q-minimizer of F if there exists Q 1 such that loc FŽ u; K QFŽ ; K for any W Ž, with K stž u loc If the above inequality is satisfied with Q then u is said a local minimizer of F In this section we shall assume that the integrand in Ž 11 deends only on x and z Under this assumtion we shall rove that local minimizers are -older continuous for all 1 and establish a local estimate of the older constant of u which will be useful in the next section where the general case will be considered Let G: be a continuous function such that for any x, y and z the following roerties hold: Ž G1 GŽ x, z Ž z gž x, z, Ž G gž x, z LŽ z, Ž Ž 3 G g x, z g y, z x y z, where g is convex in z and :,, is a continuous, not decreasing, bounded function with Ž ere,, 1 It is not restrictive, as we shall do in the sequel, to assume also 1

4 OLDE COTIUITY OF LOCAL MIIMIZES 581 If u W Ž, A we set loc GŽ u; A G x, DuŽ x Ž 1 Let us start with a simle algebraic lemma A LEMMA If 1 there exists a constant c such that for any,, Ž c c Proof For all 1 we have the elementary inequality 1 Ž from which the thesis immediately follows when Let us consider the case 1 If, the claim is obvious; otherwise we have 1 1 Ž Ž 4 Using this inequality to estimate in Ž we get 4 Ž and the thesis follows 1 POPOSITIO 3 Let G: x be a continuous function satisfying G 1, G, G 3, of class C with resect to z Let u W Ž Ž x be a minimizer of the functional w; Ž x GŽ x, Dw Dw Du Ž 3 Ž x Ž x in its Dirichlet class u W Ž Ž x, for some and u W Ž Ž x Then for any Du c Ž Du Ž x Ž x Ž 1 1Ž 1 Ž

5 58 CUPII, FUSCO, AD PETTI Proof We start by observing that since for all x the function Ž z G x, z is C then Ý ij i j i, j1 Ž D G x, z z for any x, z, Let be the minimizer in u W of G Ž w; GŽ x, Dw The function z GŽ x, z satisfies the assumtions of 5, Theorem ; hence from this result it follows that is locally Lischitz in and that the following estimate holds D c D for all This inequality, together with the minimality of, Ž G 1, and Ž G imlies D c Du Ž 4 Using Lemma and 4 we have Ž Du c D Ž c Du D Du D c Ž Du Ž c Du D Du D Ž 5

6 OLDE COTIUITY OF LOCAL MIIMIZES 583 and the last integral can be controlled, using the minimality of, as follows: G Ž u; G Ž ; GŽ x, Du GŽ x, D i i i DG x, D Du D 1 ij Ž 1 t D G x, Ž 1 t D tdu Ž Du i Di Ž Du j Dj dt 1 Ž Ž 1 t Ž 1 t D tdu Du D dt Ž c Du D Du D This inequality, together with the assumtions on G and the minimality of u and, yields Ž c Du D Du D GŽ x, Du GŽ x, Du G x, D G x, D GŽ x, Du Du Du GŽ x, D D Du D Du Du Du c Du D Du

7 584 CUPII, FUSCO, AD PETTI Finally the thesis follows from this inequality and 5 if we observe that c D Du Ž 1 1 1Ž 1 c D Du Ž 1 c Ž Ž Du 1Ž 1 Ž In the next roosition we rove an analogous result using an aroximation argument that allows us to remove the differentiability assumtion on G POPOSITIO 4 Let G: x satisfy Ž G, Ž G 1, and Ž G If, u, and u are as in Proosition 3, then for any 3 Du x c Du Du Ž x Ž 1 1Ž 1 Ž 6 Ž Proof As in the roof of Proosition 3 when the center of a ball is not indicated it is understood that the ball is centered in x Let Ž G h be the sequence of continuous functions in defined by 1 GhŽ x, z Ž w Gž x, z w dw Ž h 1 where is a ositive radially symmetric mollifier Using the same arguments as in 5, Lemma 4 it is easy to check that the functions Gh satisfy the assumtions of Proosition 3 More recisely G Ž x, C Ž h for all h and there exists a constant c 1 not deending on h such that

8 for any x, y, z, OLDE COTIUITY OF LOCAL MIIMIZES ž h 1 ž h u 1 G x, z z h ghž x, z, c h g Ž x, z cž L, u z h g Ž x, z g Ž y, z c Ž x y z h h otice that Gh converges uniformly to G on comact subsets of For every h let u be the minimizer in u W Ž h of the func- tional w G Ž x, Dw Dw Du h Then there exists C deending on L,,,, and, but not on h, such that Du C Dw Du 1 h for any w u W In articular we have that Du C Du Du 1 ; Ž 7 h hence the sequence u is bounded in W Ž h Thus, assing eventually to a subsequence, we may assume that a function u u W exists such that u u weakly in W Ž h Moreover the minimality of uh easily imlies that u is a Q-minimizer of h w Dw Du 1, so there exist 1 and c such that u W Ž h loc for any h More recisely, for any ball Ž x Ž x 1 the following inequality holds Žsee 7, Theorem 31 ž 1 h h x1 x1 Du c Du Du 1

9 586 CUPII, FUSCO, AD PETTI which, together with 7, imlies that for any ž 1 h Du cž, Du Du 1 Ž 8 Let us now rove that u u Fix and observe that the functional defined in Ž 3 is lower semicontinuous with resect to the weak toology of W emembering that Ž G h converges to G uniformly on comact subsets of, we have that for any k u ; lim inf GŽ x, Du Du Du h h h lim su GŽ x, Du h Duhk4 h h h Duhk4 lim su G Ž x, Du So, from the minimality of u, it follows that h h Du Du 1 Ž1 4 h h h u ; c lim su 1 Du Du k h lim su G Ž x, Du Du Du h Ž1 ck lim su 1 Du GŽ x, Du h h h Du Du, that together with 8 imlies u ; ck Ž1 Ž u; Finally as k and then we obtain u ; u;

10 OLDE COTIUITY OF LOCAL MIIMIZES 587 which imlies u u in, since by Ž G 1 the functional is strictly convex ow we can aly Proosition 3 on any u h; moreover, using the minimality of u and letting h, we have h Ž Du 1 lim inf Du ž h h h 1 ž h h h lim inf c Du Ž 1 Ž 1 1 c Ž Ž Du Du Du Ž 1 1Ž 1 Ž Estimating c Du Du as in the roof of Proosition 3, we obtain the thesis As a corollary of this roosition we state the following regularity result TEOEM 5 Let G be as in Proosition 4 and u W Ž loc be a local minimizer of functional G defined as in Ž 1 For any there exists a constant c, deending on L,,,,, and on the diameter of, such that if Ž x then for any Du c Du Ž x Ž x, In articular u C for any 1 loc Proof Proosition 4, alied with, imlies that for any Du c Ž Du Fixed, a standard iteration argument Žsee 6, 17 leads to the existence of two ositive constants, c for which the assertion holds if From this the result easily follows

11 588 CUPII, FUSCO, AD PETTI The following result, due in this form to Ekeland Žsee, will be used in the roof of Theorem 31 LEMMA 6 Let Ž V, d be a comlete metric sace and I: V Ž, a lower semicontinuous functional such that Gien, let u V be such that inf I is finite V I Ž u inf I V Then there exists V satisfying the following roerties: Ž i du, Ž Ž ii IŽ IŽ u, Ž iii is a minimizer of the functional w IŽ w dž, w We conclude this section by roving a higher integrability result u to Ž the boundary see also 1 LEMMA 7 Let G: x be a continuous function such Ž Ž that, for all z, z G z L z Let us consider u W q Ž Ž x for a certain q If is a minimizer of the functional G in the Dirichlet class u W Ž Ž x, then there exist r Ž, q and c deending on L,,, but not on u or, such that W r Ž Ž x and 1r 1q r q Ž ž ž Ž x Ž x D c 1 Du Proof As usual, whenever the center of a ball is not indicated it will be understood that the ball is centered in x Let us set Ž x if x, wž x ½ už x if x If x the standard Cacciooli inequality gives 1 x, 1 D c 1, Ž x1 Ž x1

12 which imlies OLDE COTIUITY OF LOCAL MIIMIZES 589 ž Ž x1 Ž x1 D c Ž 1 D, Ž 9 with Ž if Ž 1 otherwise Let us now consider Ž x 1 and x1 Let us fix s t and a cutoff function between Ž x and Ž x, with D s 1 t 1 Ž t s Observing that u on, we easily obtain u D c 1 Du Ž t s s x1 x1 Ž t Ž x 1 s Ž x 1 c D From this inequality, arguing in a standard way Žsee the roof of 7, Theorem 31, we get Ž x 1 D ž u c 1 c Du x1 x1 c 1 D Du c Du ; hence it follows that Ž x1 Ž x1 Dw c Dw c Ž 1 Du Ž x1 Ž x1 Ž x1 Ž 1 otice that Ž 1 holds not only when x and Ž x 1 1 but, by Ž 9, also if Ž x 1 Let us consider now the case of a ball such that Ž x is not emty and Ž x Fixed x in Ž x

13 59 we easily have that CUPII, FUSCO, AD PETTI Dw 3 Dw Ž x1 3Ž x ž ž 6 Ž x 6 Ž x c Dw c Ž 1 Du 8 Ž x1 8 Ž x1 c Dw c Ž 1 Du Since this estimate is true for any Ž x such that Ž x 1 8 1, it follows with an easy argument that Ž 1 holds for any Ž x 1 such that Ž x 1, ossibly with a different constant c The Gehring lemma roved in 6 yields now that if Ž x, then 1 1r 1 r Dw c Dw ž Ž x1 Ž x1 1q q Ž ž Ž x 1 c 1 Du, with suitable c and r q In articular we have roved that D c Dw c Ž 1 Du ž ž ž 1r 1 1q r q 1 1q q c D c Ž 1 Du ž ž 1 1q q c Ž 1 Du c Ž 1 Du ž ž and finally the thesis follows 3 EGULAITY OF LOCAL MIIMIZES In this section we study the regularity of local minimizers of a functional of the tye 11, where F: is a continuous function satisfying the following assumtions: for any x, y, u,, and

14 OLDE COTIUITY OF LOCAL MIIMIZES 591 z Ž 1 F F x, u, z z f x, u, z, Ž F f x, u, z L z, Ž F3 f x, u, z f y,, z x y u z, where f is convex in z and :,, is a continuous, not decreasing, bounded function with Ž As before, L,, 1 Since it is not restrictive, we shall henceforth assume to be concave We can now state our main result TEOEM 31 Let F: be a continuous function eri- fying assumtions F, F, and F aboe If u W Ž 1 3 loc is a local minimizer of the functional FŽ w; F x, wž x, DwŽ x,, then u C Ž loc for all 1 Proof Since we want to rove a local result, it is not restrictive to Ž assume that see 7 u W q Ž for some q and that for any ball Ž x 1 1q q Du c 1 Du Ž 31 Ž x1 Ž x1 1 Ž, Moreover see 7 we can assume that u C Ž for some Ž, 1 ; thus let us denote simly by u the older constant of u in Let us fix Ž x such that Ž x 4 As before we shall not indicate the center of a ball when it is x Ste 1 For any x, z we set GŽ x, z FŽ x, už x, z

15 59 CUPII, FUSCO, AD PETTI Let G denote the functional defined in Ž 1 Let be the minimizer of G in u W 1 Ž Using the minimality of u, we have GŽ u FŽ ; Ž Ž G F x, x, D x F x, u x, D x Ž G D x u x 3 Let r Ž, q be the exonent given by Lemma 7 Using the boundedness and concavity of, together with Ž 31, we can control the last integral as follows: D Ž x už x r D r rž r Ž x už x r r q q Ž c 1 Du x u x ž Ž r r c Ž x už x Ž 1 Du, Ž 33 with Ž r r ecalling the Cacciooli inequality for the minimizer u Žsee 7, we have ž ž u c D Du 4 1 ž 1 ž c D Du c 1 Du

16 OLDE COTIUITY OF LOCAL MIIMIZES u u ž ž c 1 c cu Ž c Finally this relation, together with 3, 33, and the minimality of, imlies Ž GŽ u inf G c c Ž 1 Du uw 1 Ž 4 1 Ste We argue as in 4 Let us define Ž Ž c c Ž 1 Du 4 and aly Lemma 6 to the sace V u W 1 Ž endowed with the distance 1 Ž 1 dž w, w Dw Dw 1 1 Then there exists a function u W such that 1 Ž 1 Du D Ž, GŽ GŽ u, Ž 34 1 Ž is a minimizer of G w Dw D Ž 35 The minimality of imlies that for any W GŽ ; st GŽ ; st Ž 1 Ž st Ž D D D 1 st G ; st D 1 Ž D D cž st st

17 594 CUPII, FUSCO, AD PETTI From this inequality it easily follows that is a Q-minimizer, with Q deending only on L and, of the functional Ž w Dw 1 and then Žsee 7 there exist s Ž, q and c, indeendent on, such that ž s s D c D c 1 c Ž 1 Du Ž 36 4 We remark that the function G satisfies Ž G, Ž G, and Ž G 1 3 with relaced by the function given by Ž 1 ½ Ž 5 Ž t max t u t, ct Ž 37 Alying Proosition 4 to the functional in Ž 35, with Ž Ž 1 and u, we have that for any 1 1 Du D Du D Ž c Ž Ž 1 D c Ž c Du D c Ž Ž 1 Du 1Ž 1 1Ž 1 Ž 4 c 1 Du c Du D

18 OLDE COTIUITY OF LOCAL MIIMIZES 595 Finally we have to estimate the last integral Choosing Ž, 1 such that s 1 1, using Ž 31, Ž 36, Ž 34, and Ž 37 we get Du D s Ž 1 s ž ž Du D Du D 4 c 1 Du Ž ž 1 Ž 1 Ž 4 1 c 1 Du So we have roved that if x and if then 4 ½ Ž 5 x 4 Ž x Du c 1 Du for a certain not deending on From this inequality the thesis Ž easily follows by a standard iteration argument see 6, 17 We observe that the result stated in Theorem 31 is shar in the sense that even when F deends only on x and z we cannot exect in general that local minimizers are locally Lischitz, as it is shown by the following examle, which is a suitable modification of a well known examle concerning the regularity of classical solutions of Poisson equation Žsee 9, Cha 4 EXAMPLE 3 Let D be the unit disk in We define two functions w, f: D as follows 1 k k wž x, y Ý Ž x, y xy, k k 1 k k k fž x, y Ý Ž x, y xy k k 1 ž k1 k k k k x, y y x, y x, k 1 x y

19 596 CUPII, FUSCO, AD PETTI with C Ž, 1on D, Ž x, y 1 for all Ž x, y and st D, where D is the disk of radius centered at the origin It is easy to rove that wž x, y fž x, y in the classical sense and that f is a continuous function ow let W Ž D be a weak solution of f Ž 38 x w x Since the function u x, y x, y is a distributional solution of 38 it follows that u is a distributional solution of Lalace equation; hence it is harmonic in the classical sense In articular it follows that u W Ž D ence u is a local minimizer of the functional ² : D F D x, y g x, y, D x, y dy, with g f,, which satisfies the assumtions of Theorem 31 owever u is not a Lischitz function, since lim už, t už, t t It is clear from the roof of Theorem 31 that this result can be generalized in various directions A ossible extension is rovided by the next result ere * denotes the Sobolev exonent Ž if and any number greater than 1 if TEOEM 33 Let u W be a local minimizer of the functional loc F Ž w; F x, wž x, DwŽ x h x, wž x, where h: is a Caratheodory function such that hž x, u LŽ1 u t with t *, F satisfies the assumtions of Theorem 31 and moreoer FŽ x, u, min FŽ x, u, z Ž x, u Ž 39 z, Then u C for all 1 loc Proof The roof of the result closely follows the one of Theorem 31 enceforth we shall only indicate the necessary changes Define G and as in the roof of Theorem 31 Since u is bounded, from Ž 39 we easily get by a truncation argument that is bounded too and L Ž

20 u L Ž OLDE COTIUITY OF LOCAL MIIMIZES 597 Arguing as before we obtain that Ž GŽ u inf G c c Ž 1 Du c Defining now uw 1 Ž 4 Ž Ž c c Ž 1 Du c and fixed one can now set 4 Ž 1 Ž 1 ½ 5 t max t u t, ct, With this choice the final estimate becomes ½ 5 Du c Ž Ž 1 Du c and again the result follows by the iteration argument in 6, 17, and by the arbitrary choice of 4 EFEECES 1 E Di enedetto, C 1 local regularity of weak solutions of degenerate ellitic equations, onlinear Anal 7 Ž 1983, 8785 I Ekeland, onconvex minimization roblems, ull Am Math Soc o 3 Ž 1979, L Esosito and G Mingione, Some remarks on the regularity of weak solutions of degenerate ellitic systems, e Mat Coml 1 o 1 Ž 1998, V Ferone and Fusco, Continuity roerties of minimizers of integral functionals in a limit case, J Math Anal Al Ž 1996, 75 5 I Fonseca and Fusco, egularity results for anisotroic image segmentation models, Ann Scuola orm Su Pisa 4 Ž 1997, M Giaquinta, Multile integrals in calculus of variations and nonlinear ellitic systems, in Annals of Math Studies, Vol 15, Princeton Univ Press, Princeton, J, M Giaquinta and E Giusti, Quasi-minima, Ann Inst Poincare Anal on Lineaire 1 Ž 1984, M Giaquinta and G Modica, emarks on the regularity of the minimizers of certain degenerate functionals, Manuscrita Math 57 Ž 1986, D Gilbarg and S Trudinger, Ellitic Partial Differential Equations of Second Order, Sringer-Verlag, erlinew York, E Giusti, Metodi diretti nel calcolo delle variazioni, UMI, ologna, P artman and G Stamacchia, On some nonlinear ellitic differential-functional equations, Acta Math 115 Ž 1966, J J Manfredi, egularity for minima of functionals with -growth, J Differential Equations 76 Ž 1988, L C Piccinini and S Sagnolo, On the older continuity of solutions of second order ellitic equations in two variables, Ann Scuola orm Su Pisa 6 Ž 197, 3914