Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

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724 D. Bucur / Journal of Functional Analysis 236 (2006) 712–725 there exists a subsequence of the sets {vn >tr} and a set Ωtr , such that {vnk >tr} γ -converges to Ωtr . Consequently, by a diagonal extraction procedure as in Theorem 3.1, we construct an obstacle h such that {h >tr} =Ωtr and the sequence vnk converges in the sense of obstacles to h.Fromtheγ-convergence of the level sets, we get (v − tr) + ∈ W 1,p (Ωtr 0 ), hence h v.This contradicts (13), from the Mosco convergence. It is needless to say that vn are not, in general, p-superharmonic. Remark 4.3. In [3], the authors give an example of a sequence of positive functions vn which are not superharmonic such that inequality (1) fails to be true for a suitable sequence un of superharmonic functions. This construction is done around the pioneering result of Cioranescu and Murat [10] on the “strange term” appearing in the relaxation process through the γ -convergence. The presence of the strange term is an argument of non-γ -convergence of obstacles! So, from this point of view it is not surprising that inequality (1) is violated, although the choice of un has to be done carefully. Remark 4.4. We give in the sequel an example of non-superharmonic functions vn for which the second assertion of Theorem 3.1 holds. Let η ∈ C1 (R2 , R) be a periodic function of period (l1,l2) and ϕ ∈ C∞ 0 (Ω). We consider the sequence of functions x vn = wΩ + ϕεη . ε It is clear that this sequence converges weakly but not strongly in H 1 0 (Ω) to v = wΩ, provided that ϕ or η are not the zero functions and ε ↓ 0. Inequality (1) is satisfied for all admissible sequences (un). This is a consequence of the obstacle convergence of vn towards v which can be easily proved. Nevertheless for small ε, the functions vn are not superharmonic! References [1] H. Attouch, Variational Convergence for Functions and Operators, Pitman, London, 1984. [2] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (6) (1992) 581–597. [3] M. Briane, G. Mokobodzki, F. Murat, Variations on a strange semi-continuity result, J. Funct. Anal. 227 (1) (2005) 78–112. [4] M. Briane, G. Mokobodzki, F. Murat, Semi-strong convergence of sequence satisfying a variational inequality, preprint, http://www.insa-rennes.fr/~mbriane. [5] D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Progr. in Nonlinear Differential Equations, vol. 65, Birkhäuser, Basel, 2005. [6] D. Bucur, P. Trebeschi, Shape optimisation problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A 128 (5) (1998) 945–963. [7] D. Bucur, P. Trebeschi, A new relaxation space for obstacles, Acta Appl. Math. 79 (3) (2003) 177–194. [8] D. Bucur, G. Buttazzo, P. Trebeschi, An existence result for optimal obstacles, J. Funct. Anal. 162 (1) (1999) 96– 119. [9] J. Casado Diaz, A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains, SIAM J. Math. Anal. 31 (3) (2000) 581–624. [10] D. Cioranescu, F. Murat, Un terme étrange venu d’ailleurs, in: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. II, Paris, 1979/1980, in: Res. Notes in Math., vol. 60, Pitman, Boston, MA, 1982, pp. 98–138, 389–390. [11] A. Dall’Acqua, G. Sweers, On domains for which the clamped plate system is positivity preserving, in: Partial Differential Equations and Inverse Problems, in: Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 133–144.
D. Bucur / Journal of Functional Analysis 236 (2006) 712–725 725 [12] G. Dal Maso, Some necessary and sufficient conditions for convergence of sequences of unilateral convex sets, J. Funct. Anal. 62 (1985) 119–159. [13] G. Dal Maso, Γ -convergence and μ-capacities, Ann. Sc. Norm. Super. Pisa Cl. Sci. 14 (1988) 423–464. [14] G. Dal Maso, U. Mosco, Wiener’s criterion and Γ -convergence, Appl. Math. Optim. 15 (1987) 15–63. [15] G. Dal Maso, F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Super. Pisa Cl. Sci. 24 (1997) 239–290. [16] L.I. Hedberg, Spectral synthesis in Sobolev spaces and uniqueness of solutions of Dirichlet problems, Acta Math. 147 (3–4) (1981) 237–263. [17] J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993. [18] W.P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, Berlin, 1989.
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The assumption tells us that the fo
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3.4. The zeroes of ( √ z, η) D.
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4.1. Persistence of surface waves D
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When G = η ⊗ r, we obtain Becaus
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This polynomial satisfies D. Serre
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we now have (cf. (7.8)) that H. Sch
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2.2. Definition J. Van Schaftingen
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