Nonlinear Analysis 72 (2010) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage:

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1 Nonlinear Analysis 72 (2010) Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: Local C 1 ()-minimizers versus local W 1,p ()-minimizers of nonsmooth functionals Patrick Winkert Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, Berlin, Germany a r t i c l e i n f o a b s t r a c t Article history: Received 17 December 2009 Accepted 3 February 2010 MSC: 34A60 35R70 49J52 49J53 We study not necessarily differentiable functionals of the form J(u) = 1 u p dx + 1 u p dx + j 1 (x, u)dx + j 2 (x, γ u)dσ p p with 1 < p < involving locally Lipschitz functions j 1 : R R as well as j 2 : R R. We prove that local C 1 ()-minimizers of J must be local W 1,p ()- minimizers of J Elsevier Ltd. All rights reserved. Keywords: Clarke s generalized gradient Local minimizers Nonsmooth functionals p-laplacian 1. Introduction We consider the functional J : W 1,p () R defined by J(u) = 1 u p dx + 1 u p dx + j 1 (x, u)dx + j 2 (x, γ u)dσ (1.1) p p with 1 < p <. The domain R N is supposed to be bounded with Lipschitz boundary and the nonlinearities j 1 : R R as well as j 2 : R R are measurable in the first argument and locally Lipschitz in the second one. By γ : W 1,p () L q 1 ( ) for 1 < q1 < p (p = (N 1)p/(N p) if p < N and p = + if p N), we denote the trace operator which is known to be linear, bounded and even compact. Note that J : W 1,p () R does not have to be differentiable and that it corresponds to the following elliptic inclusion p u + u p 2 u + j 1 (x, u) 0 in, u ν + j 2(x, γ u) 0 on, where p u = div( u p 2 u), 1 < p <, is the negative p-laplacian. The symbol u denotes the outward pointing ν conormal derivative associated with p and j k (x, u), k = 1, 2, stands for Clarke s generalized gradient given by j k (x, s) = {ξ R : j o k (x, s; r) ξ r, r R}. address: winkert@math.tu-berlin.de X/$ see front matter 2010 Elsevier Ltd. All rights reserved. doi: /j.na

2 P. Winkert / Nonlinear Analysis 72 (2010) The term j o k (x, s; r) denotes the generalized directional derivative of the locally Lipschitz function s j k(x, s) at s in the direction r defined by j o k (x, s; r) = lim sup y s,t 0 j k (x, y + tr) j k (x, y), t (cf. [1, Chapter 2]). It is clear that j o(x, s; r) k R because j k(x, ) is locally Lipschitz. The main goal of this paper is the comparison of local C 1 () and local W 1,p ()-minimizers. That means that if u 0 W 1,p () is a local C 1 ()-minimizer of J, then u 0 is also a local W 1,p ()-minimizer of J. This result is stated in our main Theorem 3.1. Such a result was first proven for functionals corresponding to elliptic equations with Dirichlet boundary values by Brezis and Nirenberg in [2] if p = 2. They consider potentials of the form 1 Φ(u) = 2 u 2 F(x, u), where F(x, u) = u f (x, s)ds with some Carathéodory function f : R R. An extension to the more general case 0 1 < p < can be found in the paper of García Azorero et al. in [3]. We also refer the reader to [4] if p > 2. As regards nonsmooth functionals defined on W 1,p 0 () with 2 p <, we point to the paper [5]. A very inspiring paper about local minimizers of potentials associated with nonlinear parametric Neumann problems was published by Motreanu et al. in [6]. Therein, the authors study the functional φ 0 (x) = 1 p Dx p p F 0 (z, x(z))dz, x W 1,p n () with { W 1,p n () = y W 1,p () : x } n = 0, Z where x is the outer normal derivative of u and F n 0(z, x) = x 0 f 0(z, s)ds, as well as 1 < p <. A similar result corresponding to nonsmooth functionals defined on W 1,p n () for the case 2 p < was proved in [7]. We also refer the reader to the paper in [8] for 1 < p <. A recent paper about the relationship between local C 1 ()-minimizers and local W 1,p ()-minimizers of C 1 -functionals has been treated by the author in [9]. The idea of the present paper was the generalization to the more general case of nonsmooth functionals defined on W 1,p () with 1 < p < involving boundary integrals which in general do not vanish. 2. Hypotheses We suppose the following conditions on the nonsmooth potentials j 1 : R R and j 2 : R R. (H1) (i) x j 1 (x, s) is measurable in for all s R. (ii) s j 1 (x, s) is locally Lipschitz in R for almost all x. (iii) There exists a constant c 1 > 0 such that for almost all x and for all ξ 1 j 1 (x, s) it holds that ξ 1 c 1 (1 + s q0 1 ) with 1 < q 0 < p, where p is the Sobolev critical exponent Np p if p < N, = N p + if p N. (H2) (i) x j 2 (x, s) is measurable in for all s R. (ii) s j 2 (x, s) is locally Lipschitz in R for almost all x. (iii) There exists a constant c 2 > 0 such that for almost all x and for all ξ 2 j 2 (x, s) it holds that ξ 2 c 2 (1 + s q1 1 ) with 1 < q 1 < p, where p is given by (N 1)p if p < N, p = N p + if p N. (iv) Let u W 1,p (). Then every ξ 3 j 2 (x, u) satisfies the condition ξ 3 (x 1 ) ξ 3 (x 2 ) L x 1 x 2 α, for all x 1, x 2 in with α (0, 1]. (2.1) (2.2) Remark 2.1. Note that the conditions above imply that the functional J : W 1,p () R is locally Lipschitz (see [10] or [11, p. 313]). That guarantees, in particular, that Clarke s generalized gradient s J(s) exists.

3 4300 P. Winkert / Nonlinear Analysis 72 (2010) C 1 () versus W 1,p () Our main result is the following. Theorem 3.1. Let the conditions (H1) and (H2) be satisfied. If u 0 exists r 1 > 0 such that J(u 0 ) J(u 0 + h) for all h C 1 () with h C 1 () r 1, W 1,p () is a local C 1 ()-minimizer of J, that is, there then u 0 is a local minimizer of J in W 1,p (), that is, there exists r 2 > 0 such that J(u 0 ) J(u 0 + h) for all h W 1,p () with h W 1,p () r 2. Proof. Let h C 1 () and let β > 0 small. Then we have 0 J(u 0 + βh) J(u 0 ), β which means that 0 J o (u 0 ; h) for all h C 1 (). The continuity of J o (u 0 ; ) on W 1,p () and the density of C 1 () in W 1,p () imply 0 J o (u 0 ; h) for all h W 1,p (). Hence, we get 0 J(u 0 ). The inclusion above implies the existence of h 1 L q 0() with h 1 (x) j 1 (x, u 0 (x)) and h 2 L q 1( ) with h 2 (x) j 2 (x, γ (u 0 (x))) satisfying 1/q 0 + 1/q = 0 1 as well as 1/q 1 + 1/q 1 = 1 such that u 0 p 2 u 0 ϕdx + u 0 p 2 u 0 ϕdx + h 1 ϕdx + h 2 γ ϕdσ = 0, ϕ W 1,p (). (3.1) Note that Eq. (3.1) is the weak formulation of the Neumann boundary value problem p u 0 = h 1 u 0 p 2 u 0 in, u 0 ν = h 2 on, where u 0 ν means the outward pointing conormal and p is the negative p-laplacian. The regularity results in [12, Theorem 4.1 and Remark 2.2] along with [13, Theorem 2] ensure the existence of α (0, 1) and M > 0 such that u 0 C 1,α () and u 0 C 1,α () M. (3.2) In order to prove the theorem, we argue indirectly and suppose that the theorem is not valid. Hence, for any ε > 0 there exists y ε B ε (u 0 ) such that { } J(y ε ) = min J(y) : y B ε (u 0 ) < J(u 0 ), (3.3) where B ε (u 0 ) = {y W 1,p () : y u 0 W 1,p () < ε}. More precisely, y ε solves min J(y) y B ε (u 0 ), F ε (y) := 1 ) ( y u 0 pw p 1,p() εp 0. The usage of the nonsmooth multiplier rule of Clarke in [14, Theorem 1 and Proposition 13] yields the existence of a multiplier λ ε 0 such that 0 J(y ε ) + λ ε F ε (y ε).

4 P. Winkert / Nonlinear Analysis 72 (2010) This means that we find g 1 L q 0() with g 1 (x) j 1 (x, y ε (x)) as well as g 2 L q 1( ) with g 2 (x) j 2 (x, γ (y ε (x))) to obtain y ε p 2 y ε ϕdx + y ε p 2 y ε ϕdx + g 1 ϕdx + g 2 γ ϕdσ + λ ε (y ε u 0 ) p 2 (y ε u 0 ) ϕdx + λ ε y ε u 0 p 2 (y ε u 0 )ϕdx = 0, (3.4) for all ϕ W 1,p (). Next, we have to show that y ε belongs to L () and hence to C 1,α (). Case 1: λ ε = 0 with ε (0, 1]. From (3.4) we see that y ε solves the Neumann boundary value problem p y ε = g 1 y ε p 2 y ε in, y ε ν = g 2 on, As before, the regularity results in [12,13] yield (3.2) for y ε. Case 2: 0 < λ ε 1 with ε (0, 1]. Multiplying (3.1) with λ ε and adding (3.4) yields y ε p 2 y ε ϕdx + λ ε u 0 p 2 u 0 ϕdx + λ ε (y ε u 0 ) p 2 (y ε u 0 ) ϕdx = (λ ε h 1 + g 1 + λ ε u 0 p 2 u 0 )ϕdx (λ ε y ε u 0 p 2 (y ε u 0 ) + y ε p 2 y ε )ϕdx (λ ε h 2 + g 2 )γ ϕdσ. (3.5) With (3.5) in mind, we introduce the operator T ε : R N R N given by T ε (x, ξ) = ξ p 2 ξ + λ ε H p 2 H + λ ε ξ H p 2 (ξ H), where H(x) = u 0 (x) and H (C α ()) N for some α (0, 1]. It is clear that T ε (x, ξ) C( R N, R N ). For x we have (T ε (x, ξ), ξ) R N = ξ p + λ ε ( ξ H p 2 (ξ H) H p 2 ( H), ξ H ( H)) R N ξ p for all ξ R N, (3.6) where (, ) R N is the inner product in R N. The estimate (3.6) shows that T ε satisfies a strong ellipticity condition. Hence, the equation in (3.5) is the weak formulation of the elliptic Neumann boundary value problem div T ε (x, y ε ) = (λ ε h 1 + g 1 + λ ε ( u 0 p 2 u 0 + y ε u 0 p 2 (y ε u 0 )) + y ε p 2 y ε ) in, v ε ν = (λ εh 2 + g 2 ) on. Using again the regularity results in [12] in combination with (3.6) and the growth conditions (H1)(iii) as well as (H2)(iii) proves y ε L (). Note that D ξ T ε (x, ξ) R N b 1 + b 2 ξ p 2, (3.7) where b 1, b 2 are some positive constants. We also obtain (D ξ T ε (x, ξ)y, y) R N = ξ p 2 y 2 + (p 2) ξ p 4 (ξ, y) 2 + λ R N ε ξ H p 2 y 2 + λ ε (p 2) ξ H p 4 (ξ H, y) 2 R { N ξ p 2 y 2 if p 2 (p 1) ξ p 2 y 2 if 1 < p < 2 min{1, p 1} ξ p 2 y 2. (3.8) Because of (3.7) and (3.8), the assumptions of Lieberman in [13] are satisfied and, thus, Theorem 2 in [13] ensures the existence of α (0, 1) and M > 0, both independent of ε (0, 1], such that y ε C 1,α () and y ε C 1,α () M, for all ε (0, 1]. (3.9)

5 4302 P. Winkert / Nonlinear Analysis 72 (2010) Case 3: λ ε > 1 with ε (0, 1]. Multiplying (3.1) with 1, setting v ε = y ε u 0 in (3.4) and adding these new equations yields (u 0 + v ε ) p 2 (u 0 + v ε ) ϕdx u 0 p 2 u 0 ϕdx + λ ε v ε p 2 v ε ϕdx Defining again = ( u 0 p 2 u 0 v ε + u 0 p 2 (v ε + u 0 ) λ ε v ε p 2 v ε )ϕdx + T ε (x, ξ) = 1 λ ε ( H + ξ p 2 (H + ξ) H p 2 H) + ξ p 2 ξ and rewriting (3.10) yields the equation (h 1 g 1 )ϕdx + (h 2 g 2 )γ ϕdσ. (3.10) div T ε (x, v ε ) = 1 λ ε ( u 0 p 2 u 0 v ε + u 0 p 2 (v ε + u 0 ) λ ε v ε p 2 v ε + h 1 g 1 ) in, v ε ν = 1 λ ε (h 2 g 2 ) on. As above, we have the following estimates: (T ε (x, ξ), ξ) R N ξ p for all ξ R N, (3.11) D ξ T ε (x, ξ) R N a 1 + a 2 ξ p 2, (3.12) (D ξ T ε (x, ξ)y, y) R N min{1, p 1} ξ p 2 y 2, (3.13) with some positive constants a 1, a 2. Due to (3.11) along with [12], we obtain v ε L (). The statements (3.12) as well as (3.13) allow us to apply again the regularity results of Lieberman which implies the existence of α (0, 1) and M > 0, both independent of ε (0, 1], such that (3.9) holds for v ε. Because of y ε = v ε + u 0 and (3.2), we obtain (3.9) in the case λ ε > 1. Summarizing, we have proved that y ε L () and y ε C 1,α () for all ε (0, 1] with α (0, 1). Let ε 0. We know that the embedding C 1,α () C 1 () is compact (cf. [15, p. 38] or [16, p. 11]). Hence, we find a subsequence y εn of y ε such that y εn ỹ in C 1 (). By construction we have y εn u 0 in W 1,p () which yields ỹ = u 0. So, for n sufficiently large, say n n 0, we have y εn u 0 C 1,α () r 1, which provides J(u 0 ) J(y εn ). (3.14) However, the choice of the sequence (y εn ) implies J(y εn ) < J(u 0 ), n n 0 (see (3.3)) which is a contradiction to (3.14). This completes the proof of the theorem. References [1] F.H. Clarke, Optimization and Nonsmooth Analysis, 2nd edition, in: Classics in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, [2] H. Brezis, L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (5) (1993) [3] J.P. García Azorero, I. Peral Alonso, J.J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (3) (2000) [4] Z. Guo, Z. Zhang, W 1,p versus C 1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl. 286 (1) (2003) [5] D. Motreanu, N.S. Papageorgiou, Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential, Nonlinear Anal. 56 (8) (2004) [6] D. Motreanu, V.V. Motreanu, N.S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J. 58 (2008) [7] G. Barletta, N.S. Papageorgiou, A multiplicity theorem for the Neumann p-laplacian with an asymmetric nonsmooth potential, J. Global Optim. 39 (3) (2007) [8] A. Iannizzotto, N.S. Papageorgiou, Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities, Nonlinear Anal. 70 (9) (2009) [9] P. Winkert, Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values, Adv. Differential Equations (in press). [10] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1) (1981) [11] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II, in: Mathematics and its Applications, vol. 500, Kluwer Academic Publishers, Dordrecht, 2000.

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