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. A note on splitting-type variational problems with subquadratic growth Dominic reit Saarbrücken 009

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3 Fachrichtung 6.1 Mathematik Preprint No. Universität des Saarlandes submitted: September, 009 A note on splitting-type variational problems with subquadratic growth Dominic reit Saarland University Department of Mathematics P.O. ox Saarbrücken Germany Dominic.reit@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 We consider variational problems of splitting-type, i.e. we want to minimize [f( w) + g( n w)] dx Ω where = ( 1,..., n 1 ). Thereby f and g are two C -functions which satisfy power growth conditions with exponents 1 < p q <. In case p there is a regularity theory for minimizers u : R n Ω R N without further restrictions on p and q if n = or N = 1. In the subquadratic case the results are much weaker: we get C 1,α -regularity, if we require q p + for n = or q < p + for N = 1. In this paper we show C 1,α -regularity under the bounds q < p+ resp. q <. p 1 Introduction In this paper we discuss regularity results for local minimizers u : Ω R N of variational integrals I[u, Ω] := F ( u) dx (1.1) where Ω denotes an open set in R n and where F : R nn [0, ) satisfies an anisotropic growth condition, i.e. Ω C 1 Z p c 1 F (Z) C Z q + c, Z R nn (1.) with constants C 1, C > 0, c 1, c 0 and exponents 1 < p q <. The study of such problems was pushed by Marcellini (see [Ma1] and [Ma]) and today it is a well known fact that there is no hope for regularity of minimizers if p and q differ too much (compare [Gi] and [Ho] for counter examples). Under mild smoothness conditions on F (the case of (p, q)-elliptic integrands) the best known statement is the bound q < p + (1.3) for regularity proved by ildhauer and Fuchs [F1], where one has to suppose local boundedness of minimizers. To get better results additional assumptions are necessary. Therefore we consider decomposable integrands, which means we have F (Z) = f( Z) + g(z n ) (A1) 1

6 for Z = (Z 1,..., Z n ) with Z i R N and Z = (Z 1,..., Z n 1 ). Thereby f and g are functions of class C and we assume power growth conditions: λ(1 + Z ) p X D f( Z)( X, X) Λ(1 + Z ) p X, λ ( q 1 + Z n ) X n D g(z n )(X n, X n ) Λ ( q (A) 1 + Z n ) X n for all Z = ( Z, Z n ), X = ( X, X n ) R nn with positive constants λ, Λ and exponents 1 < p q <. Assuming (A) it is easy to see, that we have a condition of the form (1.) for F. In case p ildhauer, Fuchs and Zhong show, that local minimizers u W 1,p loc L loc (Ω, RN ) of (1.1) are of class C 1,α without further assumptions on p and q, if n = or N = 1 (see [F] and [FZ]). Additionally to (A1) and (A) in case n = they have to suppose f(z 1 ) = f( Z 1 ) and g(z n ) = ĝ( Z ), (A3) with two functions f and ĝ which are strictly increasing. This is for using the maximum principle of [DLM]. In [F3] one can find partial regularity results in this topic, but they are much weaker and not independent of dimension. If we have a look at the subquadratic situation, we find strong restrictions on p and q for receiving regular solutions: q < p + for N = 1, see [F1], and q p + for n =, see [F], Remark 5. Thereby in both cases the assumption u L loc (Ω, RN ) is necessary which we can get rid of if n =. The aim of this paper is to improve the above statements for local minimizers of (1.1). Definition 1.1 We call a function u W 1,1 loc (Ω, RN ) a local minimizer of (1.1), if we have for all Ω Ω Ω F ( u) dx < and Ω F ( u) dx Ω F ( v) dx for all v W 1,1 loc (Ω, RN ), spt(u v) Ω. Our main Theorem reads as follows: THEOREM 1.1 For any local minimizer u W 1,p loc (Ω, RN ) of (1.1) with 1 < p < we have under the assumptions (A1) and (A):

7 (a) If we have n =, (A3) and q < p + p, (A) then u C 1,α (Ω, R N ) for all α < 1. (b) If N = 1 and u L loc (Ω), so one gets u C1,α (Ω) for all α < 1. Remark 1. If we have N = 1 Theorem 1.1 b) gives (together with the results from [FZ1]) C 1,α -regularity for all choices of 1 < p q <. In the D-case we additionally have (A). This hypothesis is needed for calculating the term (Γ i = 1 + i u, i = 1, ) Γ q Γ 1 dx. Note that we have an arbitrary wide range of anisotropy for p. ut for p 1 the bound is also much better than the bound q p + from [F]. In the situation n = we can get rid of the assumption u L loc (Ω, RN ), see [i] (section ) for details. Under suitable conditions on D x D e P f and D xd Pn g it is possible to extend our result to the non-autonomous situation, which means densities F = F (x, Z) and splitting-type integrands (compare [F], Remark 3 and [FZ1], Remark 1.). C 1,α -regularity for n = From now on we assume the conditions of Theorem 1.1 a). Let u W 1,p loc (Ω, RN ) be a local minimizer of (1.1) and fix x 0 Ω. Now it is possible to find a radius R > 0 such that u L loc ( R(x 0 ), R N ) (compare [i], section, for details). From (A3) and the maximum-principle of [DLM] we get u L loc ( R(x 0 ), R N ). For 0 < ɛ 1 (u) ɛ denotes the mollification of u with radius ɛ (see [Ad]). Now we choose R 0 < R and get sup ɛ>0 (u) ɛ <. For a fixed q > max {q, } let δ := δ(ɛ) := ɛ 1 + ( u) ɛ eq L eq () and F δ (Z) := δ ( 1 + Z ) eq + F (Z) 3

8 for Z R nn. With := R0 (x 0 ) we define u δ as the unique minimizer of I δ [w, ] := F δ ( w)dx (.1) in (u) ɛ + W 1,eq 0 (, R N ). Some elementary properties of u δ are summarized in the following Lemma (see [F], Lemma 1, for further references): Lemma.1 We have as ɛ 0: u δ u in W 1,p (, R N ), ( δ 1 + uδ ) eq dx 0 and F ( u δ )dx F ( u)dx. sup δ>0 u δ L () <. u δ W 1, loc L loc (Ω, RN ). We need the following Caccioppoli-type inequality which is standard to proof: Lemma. For η C0 (), arbitrary γ {1,..., n} and Q R nn we have η D F δ ( u δ )( γ u δ, γ u δ )dx c D F δ ( u δ )([ γ u δ Q γ ] η, [ γ u δ Q γ ] η)dx for a constant c > 0 independent of δ. Analogous to [F] we must prove the following statement for H δ which is defined by with sum over γ {1, }: H δ := D F δ ( u δ )( γ u δ, γ u δ ) Lemma.3 We have H δ L loc () uniform in ɛ and u δ W 1,t loc () uniform in ɛ for all t <. Proof: We consider for Γ i,δ := 1 + i u δ, i {1, }, f 1 (ρ) := Γ p+ dx ρ and f (ρ) := Γ q,δ dx ρ

9 separately. Let η C0 ( ) with 0 η 1, η 1 on ρ and η c/(r ρ) 1. Following [F] we see [ ] f 1 (ρ) c 1 + η ηγ p+1 dx + η Γ p 1 1 u δ dx for a constant c independent of ρ, r and δ using uniform bounds on u δ. y Young s inequality we get for a suitable β > 0 the upper bound c(τ)(r ρ) β + τ Γ p+ dx for the first term on the r.h.s. (τ > 0 is arbitrary). For the second one we obtain by (A) η Γ p 1 1 u δ dx c(τ) η Γ p 1 1 u δ dx + τ η Γ p+ dx c(τ) η Hδ dx + τ η Γ p+ dx. As a consequence f 1 (ρ) c(τ) η Hδ dx + c(τ)(r ρ) β + τ Γ p+ dx. (.) For f (ρ) we receive (following ideas of [F5]) by Sobolev s inequality ( ) f (ρ) = Γ q,δ dx ηγ q,δ dx ρ [ c η Γ q,δ dx + ηγ q 1,δ u δ dx]. Using Lemma.1, we get [ f (ρ) dx c(r ρ) 1 + c ηγ q 1,δ u δ dx]. From Hölder s inequality we deduce [...] c Γ q,δ dx η Γ q,δ u δ dx r c η Hδ dx 5

10 by Lemma.1, part 1. enough we receive ) (Γ p+ + Γ q,δ ρ Combining this with (.) and choosing τ small dx c(r ρ) β + c η Hδ dx + 1 From Lemma. we deduce for Q = 0 η Hδ dx c D F δ ( u δ )( 1 u δ η, 1 u δ η) dx r +c D F δ ( u δ )( u δ η, u δ η) dx =: c [J 1 + J ]. Thus we have by (A) J c η Γ q,δ Γ,δ dx + c η Γ p Γ,δ dx r +cδ η Γ eq δ Γ,δ dx r c η Γ q,δ dx + c η Γ,δ dx r +cδ η Γ eq δ dx c(r ρ), Γ p+ dx. (.3) if we note Lemma.1, part 1, and p q. Examining J 1 one sees J 1 c η Γ q,δ Γ dx + c η Γ p Γ dx r +cδ η Γ eq δ Γ dx c(r ρ) + c η Γ q,δ Γ dx. (.) Considerating the last critical term, one can follow by Young s inequality (τ > 0 is arbitrary) c η Γ q,δ Γ dx τ Γ p+ dx + c(τ ) η p+ q p+ p Γ p,δ dx. (A) gives q p + p 6 < q.

11 We deduce from Young s inequality η Γ q,δ Γ dx c(τ )(r ρ) β + τ Γ p+ dx + τ Γ q,δ dx. r (.5) Now we combine (.) and (.5) and get by a suitable choice of τ c η Hδ dx c(r ρ) β + 1 Γ p+ dx + 1 Γ q,δ dx. (.6) Inserting this into (.3) we get ) (Γ p+ + Γ q,δ dx c(r ρ) β + 1 ρ ) (Γ p+ + Γ q,δ dx. for all ρ < r R < R 0 with c = c(r ). From [Gi] (Lemma 5.1, S. 81) we deduce uniform boundedness of 1 u δ in L p+ loc (, RN ) and u δ in L q loc (, RN ), as well as weak convergence of subsequences in these spaces. So we get 1 u L p+ loc (Ω, RN ) and u L q loc (Ω, RN ). y this result we can infer from (.6) uniform boundedness of H δ in L loc (). Since Γ p Γ p 1 u δ ch δ, Γ q,δ Γ q,δ u δ ch δ we obtain (by Lemma.1) uniform boundedness of Γ p and Γ q,δ in W 1, and so we have arbitrary high uniform integrability of 1 u δ and u δ. loc () Now we define h := Γ p, h,δ := Γ q,δ, h 3,δ := δγ eq δ and h δ := ( h + h,δ + h 3,δ ) 1. We see following [F] for (z 0 ) (z 0 ) Hδ dx c (z 0 ) 1 s (H δ h δ ) s dx (z 0 ) u δ s dx 1 s, (.7) where... denotes the mean value. Note h := Γ p Γ p on account of p <. y definition of h δ, H δ and (A) we receive u δ chδ h δ 7

12 and thereby Hδ dx 1 c 1 s (H δ h δ ) s dx, (.8) (z 0 ) (z 0 ) which is exactly (30) in [F]. To use further arguments of [F], let h := Γ p, h,δ := Γ q,δ, h3,δ := δγ eq δ and h δ := ( h + h,δ + h ) 1 3,δ. For κ := min {p/( p), q/(q ), q/( q )} > 1 (note p > 1 and q >, if q we have a range between p and q, small enough to quote the results of [F]) we have h κ δ c h δ and thereby h δ µ h δ + c µ Now one can end up the proof as in [F]. for all µ > 0. 3 C 1,α -regularity for N = 1 In this section we work with the Hilbert Haar-regularization (see [FZ]): Let := R (x 0 ) Ω fixed, then we define u ɛ as the unique minimizer of I[, ] in the space of Lipschitz-functions R on boundary data (u) ɛ (see [MM], Thm., p. 16), which denotes the mollification of u. So we can quote (compare [FZ], p., and [MM], Thm. 5, p. 16) Lemma 3.1 We have as ɛ 0: u ɛ u in W 1,p (), F ( u ɛ )dx F ( u)dx; sup ɛ>0 u ɛ L () < ; u ɛ C 1,µ () W, loc () for all µ < 1. 8

13 With these preparations, ildhauer, Fuchs und Zhong show sup u ɛ L t ( ρ(x 0 )) < (3.1) ɛ>0 for all t < and all ρ < R (see [FZ]). W.l.o.g. we assume p q. In this case we have (compare (A)) λ(1 + Z ) p X D F (Z)(X, X) Λ(1 + Z ) q X. (3.) Now we can reproduce the proof of [i], Thm 5.. Let Γ ɛ := 1 + u ɛ and τ(k, r) := Γ q ɛ (Γ ɛ k) dx A(k,r) with A(k, r) := [Γ ɛ > k]. y arguments from [i] one can show τ(h, r) ( r r) n 1 n 1 c s (h k) n 1 1 n 1 s t τ(k, r) 1 n 1 n 1 s[1+ 1 t ] for 0 < k < h and 0 < r < r < R. Here s, t > 1 are chosen such that [ 1 n ] > 1 n 1 s t and c is independent of h, k, r, r and ɛ. If we use [St], Lemma 5.1, we get u in L loc (, Rn ) (see [i], p. 66, for details). According to the standard theory for elliptic equations or variational problems with standard growth conditions (compare [Gi]) we can follow the claim of Theorem 1.1. References [Ad] [i] [i] R. A. Adams (1975): Sobolev spaces. Academic Press, New York- San Francisco-London. M. ildhauer (003): Convex variational problems: linear, nearly linear and anisotropic growth conditions. Lecture Notes in Mathematics 1818, Springer, erlin-heidelberg-new York. M. ildhauer (003): Two-dimensional variational problems with linear growth. Manus. Math. 110, [r] D. reit (009): Regularitätssätze für Variationsprobleme mit anisotropen Wachstumsbedingungen. PhD thesis, Saarland University. 9

14 [F1] [F] [F3] [F] [F5] [FZ] [DLM] [Gi] [Gi] [Ho] [Ma1] M. ildhauer, M. Fuchs (00): Elliptic variational problems with nonstandard growth. International Mathematical Series, Vol. I, Nonlinear problems in mathematical physics and related topcs I, in honor of Prof. O.A. Ladyzhenskaya. y Tamara Rozhkovska, Novosibirsk, Russia (in Russian), 9-6. y Kluwer/Plenum Publishers (in English), M. ildhauer, M. Fuchs (007): Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals. Manus. Math. 13, M. ildhauer, M. Fuchs (007): Partial regularity for minimizers of splitting-type variational integrals. Preprint 195, Universität des Saarlandes. M. ildhauer, M. Fuchs (003): Twodimensional anisotropic variational problems. Calcules of Variations Variations 16, M. ildhauer, M. Fuchs (009): Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in D with applications to nonlinear Hencky-materials. Preprint 3, Universität des Saarlandes. M. ildhauer, M. Fuchs, X. Zhong (007): A regularity theory for scalar local minimizers of splitting-type variational integrals. Ann. SNS Pisa VI(5), A. D Ottavio, F. Leonetti, C. Mussciano (1998): Maximum prinziple for vector valued mappings minimizing variational integrals. Atti Sem. Mat. Fis. Uni. Modena XLVI, M. Giaquinta (1987): Growth conditons and regularity, a counterexample. Manus. Math. 59, 5-8. M. Giaquinta (1983): Mulitiple integrals in the calcules of variations an nonlinear elliptic systems. Ann. Math. Studies 105, Princeton University Press, Princeton. M. C. Hong (199): Some remarks on the minimizers of variational integrals with non standard growth conditions. oll. U.M.I. (7) 6- A, P. Marcellini (1989): Regularity of minimizers of integrals in the calcules of variations with non standard growth condtions. Arch. Rat. Mech. Anal. 105,

15 [Ma] [MM] [St] P. Marcellini (1991): Regularity and existence of elliptic equations with (p,q)-growth conditions. Journal of Differential Equations 90, U. Massari, M. Miranda (198): Minimal surfaces of codimension one. North-Holland Math. Stud. 91, North-Holland, Amsterdam. G. Stampacchia (1965): Le problème de Dirichlet pour les equations elliptiques du second ordre à coefficients dicontinus. Ann Inst. Fourier Grenoble 15.1,