Universität des Saarlandes. Fachrichtung 6.1 Mathematik

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1 Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 133 Higher order variational problems on two-dimensional domains Michael Bildhauer and Martin Fuchs Saarbrücken 005

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3 Fachrichtung 6.1 Mathematik Preprint No. 133 Universität des Saarlandes submitted: 1 st of March, 005 Higher order variational problems on two-dimensional domains Michael Bildhauer Saarland University Dep. of Mathematics P.O. Box D-6601 Saarbrücken Germany bibi@math.uni-sb.de Martin Fuchs Saarland University Dep. of Mathematics P.O. Box D-6601 Saarbrücken Germany fuchs@math.uni-sb.de

4 Edited by FR 6.1 Mathematik Universität des Saarlandes Postfach Saarbrücken Germany Fax: preprint@math.uni-sb.de WWW:

5 Abstract Let u: R R M denote a local minimizer of J[w] = f( k w) dx, where k and k w is the tensor of all k th order (weak) partial derivatives. Assuming rather general growth and ellipticity conditions for f, we prove that u actually belongs to the class C k,α (; R M ) by the way extending the result of [BF] to the higher order case by using different methods. A major tool is a lemma on the higher integrability of functions established in [BFZ]. 1 Introduction Let denote a bounded domain in R and consider a function u: R M which locally minimizes the variational integral J[w, ] = f( k w) dx, where k w represents the tensor of all k th order (weak) partial derivatives. Our main concern is the investigation of the smoothness properties of such local minimizers under suitable assumptions on the energy density f. For the first order case (i.e. k = 1) we have rather general results which can be found for example in the textbooks of Morrey [Mo], Ladyzhenskaya and Ural tseva [LU], Gilbarg and Trudinger [GT] or Giaquinta [Gi], for an update of the history including recent contributions we refer to [Bi]. In order to keep our exposition simple (and only for this reason) we consider the scalar case (i.e. M = 1) and restrict ourselves to variational problems involving the second (generalized) derivative. Then our variational problem is related to the theory of plates: one may think of u: R as the displacement in vertical direction from the flat state of an elastic plate. The classical case of a potential f with quadratic growth is discussed in the monographs of Ciarlet and Rabier [CR], Necǎs and Hlávácek [NH], Chudinovich and Constanda [CC] or Friedman [Fr], further references are contained in Zeidler s book [Ze]. We also like to remark that plates with other hardening laws (logarithmic and power growth case) together with an additional obstacle have been studied in the papers [BF1] and [FLM] but not with optimal regularity results. The purpose of this note is to present a rather satisfying regularity theory for a quite large of potentials allowing even anisotropic growth. To be precise let M denote the space of all ( )-matrices and suppose that we are given a function f: M [0, ) of class C which satisfies with exponents 1 < p q < the anisotropic ellipticity estimate λ(1 + ξ ) p σ D f(ξ)(σ, σ) Λ(1 + ξ ) q σ (1.1) for all ξ, σ M with positive constants λ, Λ. Note that (1.1) implies the growth condition a ξ p b f(ξ) A ξ q + B (1.) AMS Subject Classification: 9N60, 7K0 Keywords: variational problems of higher order, regularity of minimizers, nonstandard growth 1

6 with suitable constants a, A > 0, b, B 0. Let J[w, ] = f( w) dx, w = ( α β w) 1 α,β. We say that a function u W p,loc () is a local J-minimizer iff J[u, ] < for any subdomain and J[u, ] J[v, ] for all v Wp,loc () such that u v W p ( ) (here Wp,loc k () etc. denote the standard Sobolev spaces, see [Ad]). Note that (1.1) implies the strict convexity of f. Therefore, given a function u 0 Wq (), the direct method ensures the existence of a unique J- minimizer u in the class { v W p () : J[v, ] <, v u 0 W p () } which motivates the discussion of local J-minimizers. Our main result reads as follows: THEOREM 1.1 Let u denote a local J-minimizer under condition (1.1). Assume further that q < min(p, p + ) (1.3) holds. Then u is of class C,α () for any 0 < α < 1. REMARK 1.1 i) Clearly the result of Theorem 1.1 extends to local minimizers of the variational integral I[w, ] = f( w) dx + g( w) dx, where f is as before and where g denotes a density of class C satisfying 0 D g(ξ)(η, η) c(1 + ξ ) s η for some suitable exponent s. In case p any finite number is admissible for s, in case p < we require the bound s p/( p). The details are left to the reader. ii) W.l.o.g. we may assume that q : if (1.1) holds with some exponent q <, then of course (1.1) is true with q replaced by q := and (1.3) continues to hold for the new exponent. iii) If we consider the higher order variational integral f( k w) dx with k and f satisfying (1.1), then (1.3) implies that local minimizers u Wp,loc k () actually belong to the space C k,α (). iv) The degree of smoothness of u can be improved by standard arguments provided f is sufficiently regular. v) A typical example of an energy J satisfying the assumptions of Theorem 1.1 is given by J[w, ] = w dx + (1 + 1 w ) q dx with some exponent q (, ).

7 vi) Our arguments can easily be adjusted to prove C k,α -regularity of local minimizers u W k p(x),loc () of the energy (1 + k w ) p(x)/ dx provided that 1 < p p(x) < p < for some numbers p, p and if p(x) is sufficiently smooth. Another possible extension concerns the logarithmic case, i.e. we now consider the variational integral k w ln(1 + k w ) dx and its local minimizers which have to be taken from the corresponding higher order Orlicz-Sobolev space. The proof of Theorem 1.1 is organized as follows: we first introduce some suitable regularization and then prove the existence of higher order weak derivatives for this approximating sequence in Step. Here we also derive a Caccioppoli-type inequality using difference quotient methods. In a third step we deduce uniform higher integrability of the second generalized derivatives for any finite exponent. From this together with a lemma established in [BFZ] we finally obtain our regularity result in the last two steps. Proof of Theorem 1.1 Step 1. Approximation Let us fix some open domains 1 and denote by ū m the mollification of u with radius 1/m, in particular m ū m u W p ( ) 0. Jensen s inequality implies J[ū m, ] J[u, ] + τ m, where τ m 0 as m. This, together with the lower semicontinuity of the functional J, shows that J[ū m, ] m J[u, ]. (.1) Next let ρ m := ū m u W p ( ) which obviously tends to 0 as m. regularized functional J m [w, ] := ρ m [ (1 + ū m ) q dx ] 1, With these preliminaries we introduce the (1 + w ) q dx + J[w, ] and the corresponding regularizing sequence {u m } as the sequence of the unique solutions to the problems By (.1) and (.) we have J m [, ] min in ū m + W q ( ). (.) J m [u m, ] J m [ū m, ] = ū m u W p ( ) + J[ū m, ] m J[u, ], 3

8 hence one gets On account of (.3) and the growth of f we may assume Moreover, lower semicontinuity gives lim sup J m [u m, ] J[u, ]. (.3) m u m m : û in W p (). J[û, ] lim inf m J[u m, ], which together with (.3) and the strict convexity of f implies û = u (here we also note that û u W p ( )). Summarizing the results it is shown up to now that (as m ) u m u in W p (), J m [u m, ] J[u, ]. (.) Step. Existence of higher order weak derivatives In this second step we will prove that (f m (ξ) := ρ m (1 + ξ ) q/ + f(ξ)) η 6 D f m ( u m )( α u m, α u m ) dx c( η + η ) D f m ( u m ) [ u m + u m ] dx, (.5) where η C 0 ( ), 0 η 1, η 1 on 1 and where we take the sum over repeated indices. To this purpose let us recall the Euler equation Df m ( u m ) : ϕ = 0 for all ϕ W q ( ). (.6) If h denotes the difference quotient in the coordinate direction e α, α = 1,, then the test function h (η 6 h u m ) is admissible in (.6) with the result h {Df m ( u m )} : (η 6 h u m ) dx = 0. (.7) Now denote by B x the bilinear form and observe that B x = 1 0 D f m ( u m (x) + th ( h u m )(x)) dt, h {Df m ( u m )}(x) = 1 h d dt Df m( u m (x) + t[ u m (x + he α ) u m (x)]) dt d dt Df m( u m (x) + ht ( h u m )(x)) dt = 1 h 0 = B x ( ( h u m )(x), ),

9 hence (.7) can be written as B x ( ( h u m ), (η 6 h u m )) dx = 0, which means that we have η 6 B x ( ( h u m ), ( h u m )) dx = B x ( ( h u m ), η 6 h u m ) dx B x ( ( h u m ), η 6 ( h u m )) dx =: T 1 T. (.8) To handle T 1 we just observe α β η 6 = 30 α η β ηη + 6 α β ηη 5, for T we use η 6 = 6η 5 η. The Cauchy-Schwarz inequality for the bilinear form B x implies T = 6 B x (η 3 ( h u m ), η η ( h u m )) dx [ 6 [ B x ( ( h u m ), ( h u m ))η 6 dx B x ( η ( h u m ), η ( h u m ))η dx an analogous estimate being valid for T 1. Absorbing terms, (.8) turns into η 6 B x ( ( h u m ), ( h u m )) dx c( η + η ) B x ( ( h u m ) + h u m ) dx. (.9) Next we estimate (note that in the following calculations we always assume w.l.o.g. q, compare Remark 1.1, ii)) for h sufficiently small B x ( h u m ) dx c [ + (1 + u m + h ( h u m ) ) q ( h u m ) dx ( h u m ) q dx (1 + u m + h ( h u m ) ) q dx ] c (1 + u m ) q dx., 5

10 In a similar way we estimate B x h u m dx and end up with lim sup η 6 B x ( ( h u m ), ( h u m )) dx h 0 c( η + η ) (1 + u m + u m ) q dx. (.10) Since q is assumed, (.10) implies that u m W 1,loc () and h ( u m ) h 0 α ( u m ) REMARK.1 With (.10) we have and, as a consequence, in L loc () and a.e. h {Df m ( u m )} q q 1 L 1 loc ( ) uniformly w.r.t. h, Df m ( u m ) W 1 q/(q 1),loc (). This follows exactly as outlined in the calculations after (3.1) of [BF3]. With the above convergences and Fatou s lemma we find the lower bound η 6 D f m ( u m )( α u m, α u m ) dx for the l.h.s. of (.10) which gives using (1.1) η 6 (1 + u m ) p 3 u m dx c( η + η ) (1 + u m + u m ) q dx <, in particular But (.11) implies h m L r loc () for any r <, i.e. h m := (1 + u m ) p W 1,loc ( ). (.11) u m L t loc () for any t <. (.1) Using Fatou s lemma again we obtain from (.8) η 6 D f m ( u m )( α u m, α u m ) dx lim inf η 6 h {Df m ( u m )} : ( h u m ) dx h 0 = lim inf h {Df m ( u m )} : [ η 6 h u m + η 6 ( h u m )] dx (.13) h 0 On account of (.1), Remark.1 and Vitali s convergence theorem we may pass to the limit h 0 on the r.h.s. of (.13) and obtain η 6 D f m ( u m )( α u m, α u m ) dx D f m ( u m )( α u m, η 6 α u m + η 6 α u m ) dx. 6

11 This immediately gives (.5) by repeating the calculations leading from (.8) to (.9). Step 3. Uniform higher integrability of u m Let χ denote any real number satisfying χ > p/(p q), moreover we set α = χp/. For all discs B r B R any η C0 (B R ), η 1 on B r, k η c/(r r) k, k = 1,, we have by Sobolev s inequality (1 + u m ) α dx (η 3 h m ) χ dx B r B R c [ B R (η 3 h m ) t dx where t (1, ) satisfies χ = t/( t). Hölder s inequality implies [ ] χ (1 + u m ) α dx c(r, R) (η 3 h m ) dx B r B R [ ] χ c(r, R) η 6 h m dx + η 3 h m dx. B R Observing that obviously η 3 h m dx c(r, R) and that by (.5) η 6 h m dx c(r, R) B R [ c(r, R) we deduce B r (1 + u m ) α dx c(r, R) [ ] χ t (1 + u m ) p dx ] (1 + u m ) q [ u m + u m dx (1 + u m ) q dx + u m q dx χ (1 + u m ) q dx + u m dx] q, (.1) where c(r, R) = c(r r) β for some suitable β > 0. For discussing (.1) we first note that the term u m q dx causes no problems. In fact, since u m W p ( ) c < we know that u m L t loc () for any t < in case p. If p <, then we have local L t -integrability of u m provided that t < p/( p), but q < p < p/( p) on account of (1.3). As a consequence, we may argue exactly as in [ELM] or [Bi], p. 60, to derive from (.1) by interpolation and hole-filling (here q < p enters in an essential way) u m L t loc () for any t < and uniformly w.r.t. m. (.15) Note that (.15) implies with Step the uniform bound η 6 D f m ( u m )( α u m, α u m ) dx c(η) <, (.16) 7, ],

12 in particular (.16) shows h m W 1,loc( ) uniformly w.r.t. m. (.17) REMARK. i) If u is a local J-minimizer subject to an additional constraint of the form u ψ a.e. on for a sufficiently regular function ψ: R, then it is an easy exercise to adjust the technique used in [BF1] to the present situation which means that we still have (.15) so that (recall (.)) u Wt,loc () for any t <, hence u C 1,α () for all 0 < α < 1. In [Fr], Theorem 10.6, p. 98, it is shown for the special case f(w) = w that actually u C () is true, and it would be interesting to see if this result also holds for the energy densities discussed here. ii) We remark that the proof of (.15) just needs the inequality q < p, whereas the additional assumption q < p + enters in the next step. Step. C -regularity Now we consider an arbitrary disc 1 and η C0 () satisfying η 1 on B R and η c/r, η c/r. Moreover we denote by T R the annulus T R := B R and by P m a polynomial function of degree less than or equal to. Exactly as in Step (replacing u m by u m P m ) we obtain η 6 D f m ( u m )( α u m, α u m ) dx D f m ( u m )( α u m, η 6 α [u m P m ] + η 6 α (u m P m )) dx. T R With the notation [ H m := D f m ( u m )( α u m, α u m ), σ m := Df m ( u m ) we therefore have η 6 Hm dx c σ m [ η 6 u m P m + η 6 u m P m ] dx. T R Moreover, by the Cauchy-Schwarz inequality and (1.1) σ m H m [D f m ( u m )( α σ m, α σ m ) where Γ m := 1 + u m. Finally we let and obtain h m := max [Γ q m ], Γ p m σ m ch m Γ q m ch m hm, H m σ m Γ q m, hence [ ] η 6 Hm dx c H m hm η 6 u m P m + η 6 u m P m dx. (.18) T R 8

13 Letting γ = /3 we discuss the r.h.s. of (.18): H m hm η 6 u m P m dx T R [ [ c γ (H m hm ) γ dx u m P m dx R Next the choice of P m is made more precise by the requirement P m = u m dx. (.19) Then Sobolev-Poincaré s inequality together with the definition of h m gives hence [ u m P m dx c [ 3 u m γ dx T R H m hm η 6 u m P m dx c R [ γ c [ (H m hm ) γ dx (H m hm ) γ dx. ] γ γ. (.0) To handle the remaining term on the r.h.s. of (.18) we need in addition to (.19) ( u m P m ) dx = 0, which can be achieved by adjusting the linear part of P m. Then we have by Poincaré s inequality H m hm η 6 u m P m dx c R [ (H m hm ) γ dx c R [ (H m hm ) γ dx γ [ γ [ u m P m dx u m P m dx and the r.h.s. is bounded by the r.h.s. of (.0). Hence, recalling (.18) and (.0), we have established the inequality [ B R H m dx ] γ c (H m hm ) γ dx. (.1) Given this starting inequality we like to apply the following lemma which is proved in [BFZ]. 9,,

14 LEMMA.1 Let d > 1, β > 0 be two constants. With a slight abuse of notation let f, g, h now denote any non-negative functions in R n satisfying f L d loc (), exp(βgd ) L 1 loc (), h Ld loc (). Suppose that there is a constant C > 0 such that [ f d dx d C fg dx + C[ h d dx d B B B holds for all balls B = B r (x) with B = B r (x). Then there is a real number c 0 = c 0 (n, d, C) such that if h d log c0β (e + h) L 1 loc (), then the same is true for f. Moreover, for all balls B as above we have [ f d log c 0β f ] ] e + dx c[ exp(βg d ) dx][ f d dx f d,b B B +c h d log c 0β where c = c(n, d, β, C) > 0 and f d,b = ( B f d dx) 1/d. B [ e + B h ] dx, f d,b The appropriate choices in the setting at hand are d = /γ = 3/, f = H γ m, g = h γ m, h 0. We claim that exp( h mβ) dx c and Hm dx c for a constant being uniform in m. The uniform bound of the second integral follows from (.16), thus let us discuss the first one. By (.17) and Trudinger s inequality (see e.g. Theorem 7.15 of [GT]) we know that for any disc B ρ 1 exp(β 0 h m ) dx c(ρ) <, B ρ where β 0 just depends on the uniformly bounded quantities h m W 1 ( 1 ). This implies for any β > 0 and κ (0, 1) exp(βh κ m ) dx c(ρ, β, κ) <. B ρ Moreover, on account of q < p + we have Γ q m h κ m and clearly Γ p m h κ m for κ sufficiently small, which gives our claim and we may indeed apply the lemma with the result B ρ H m log c0β (e + H m ) dx c(β, ρ) < 10

15 for all discs B ρ 1 and all β > 0. Thus we have established the counterparts of (.7) and (.10) in [BFZ], and exactly the same arguments as given there lead to (.11) from [BFZ], thus we deduce the uniform continuity of the sequence {σ m } (see again [BFZ], end of Section ), hence we have uniform convergence σ m : σ for some continuous tensor σ. In order to identify σ with Df( u), we recall the weak convergence stated in(.) and also observe that u m u a.e. which can be deduced along the same lines as in Lemma.5 c) of [BF3], we also refer to Proposition 3.9 iii) of [Bi]. Therefore Df( u) is a continuous function, i.e. u is of class C 0, and finally u C () follows. Step 5. C,α -regularity of u To finish the proof of Theorem 1.1 we observe that with Step we get from (.5) the estimate 3 u m dx c( 1 ) <, 1 in particular one has for α = 1, U := α u W,loc (). Moreover we have D f m ( u m )( α u m, ϕ) dx = 0 for any ϕ C0 (). Together with the convergences (as m ) D f m ( u m ) D f( u) in L loc (), α u m U in L loc () we therefore arrive at the limit equation D f( u)( U, ϕ) dx = 0, hence U is a weak solution of an equation with continous coefficients and u C,α () for any 0 < α < 1 follows from [GM], Theorem.1. References [Ad] [Bi] Adams, R.A., Sobolev spaces. Academic Press, New York-San Francisco-London, Bildhauer, M., Convex variational problems: linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, Berlin- Heidelberg-New York, 003. [BF1] Bildhauer, M., Fuchs, M., Higher order variational inequalities with nonstandard growth conditions in dimension two: plates with obstacles. Ann. Acad. Sci. Fenn. Math. 6 (001),

16 [BF] Bildhauer, M., Fuchs, M., Two-dimensional anisotropic variational problems. Calc. Variations. 16 (003), [BF3] Bildhauer, M., Fuchs, M., Variants of the Stokes problem: the case of anisotropic potentials. J. Math. Fluid Mech. 5 (003), [BFZ] Bildhauer, M., Fuchs, M., Zhong, X., A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids. Manus. Math. 116 (005), [CC] [CR] Chudinovich, I., Constanda, C., Variational and potential methods in the theory of bending of plates with transverse shear deformation. Chapman and Hall, 000. Ciarlet, P., Rabier, P., Les équations de von Kármán. Lecture Notes in Mathematics 86, Springer, Berlin-Heidelberg-New York, [ELM] Esposito, L., Leonetti, F., Mingione, G., Regularity results for minimizers of irregular integrals with (p,q)-growth. Forum Math. 1 (00), 5 7. [Fr] Friedman, A., Variational principles and free boundary problems. Wiley- Interscience, 198. [FLM] Fuchs, M., Li, G., Martio, O., Second order obstacle problems for vectorial functions and integrands with subquadratic growth. Ann. Acad. Sci. Fenn. Math. 3 (1998), [Gi] [GM] [GT] [LU] [Mo] [NH] [Ze] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Studies 105, Princeton University Press, Princeton Giaquinta, M., Modica, G., Regularity results for some classes of higher order non linear elliptic systems. J. Reine Aangew. Math. 311/31 (1979), Gilbarg, D., Trudinger, N., Elliptic partial differential equations of the second order. Second Edition, Springer-Verlag, Berlin-Heidelberg, Ladyzhenskaya, O.A., Ural tseva, N.N., Linear and quasilinear elliptic equations. Nauka, Moskow, 196. English translation: Academic Press, New York Morrey, C. B., Multiple integrals in the calculus of variations. Grundlehren der math. Wiss. in Einzeldarstellungen 130, Springer, Berlin-Heidelberg-New York Necǎs, J., Hlávácek, I., Mathematical theory of elastic and elasto-plastic bodies. Elsevier, New York, Zeidler, E., Nonlinear functional analysis and its applications IV. Applications to mathematical physics. Springer, Berlin-Heidelberg-New York,