Remarks on perturbated systems with critical growth. (English) Zbl 0786.35061
The author considers the question on convergence almost everywhere for gradients \(Du_ j\) of a bounded sequence \(u_ j\) in the Sobolev space \(W_ 0^{1,p} (\Omega;\mathbb{R}^ N)\) which appears in the study of the Dirichlet problem for the elliptic system with critical growth perturbation:
\[
A(u)+G(u)=f,\quad u |_{\partial \Omega} =0
\]
where \(\Omega \subset \mathbb{R}^ n\) is a bounded domain, \(A(u)\) is a quasilinear operator and the perturbation \(G(u)\) is dependent on the (vector-valued) function \(u\) and its gradient \(Du\). The growth of the perturbation is called critical if it can be controlled by \(| Du |^ p\) only, where \(p\) is the same exponent for which \(A(u)\) defines a coercive mapping from \(W_ 0^{1,p}(\Omega;\mathbb{R}^ N)\) to its dual space \((W_ 0^{1,p}(\Omega;\mathbb{R}^ N))^*\).
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |
Keywords:
monotone operators; Young measures; maximal function method; biting lemma; convergence almost everywhere; Sobolev space; elliptic system; critical growth perturbationReferences:
| [1] | Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Archs ration. Mech. Analysis, 86, 125-145 (1984) · Zbl 0565.49010 |
| [2] | Adams, R. M., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 |
| [3] | Balder, E. J., A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22, 570-597 (1984) · Zbl 0549.49005 |
| [5] | Ball, J. M., A version of the fundamental theorem of Young measures, (Serre, D., Proc. Conf. Partial Diff. Eqns Contin. Models Phase Transit. (1988), Springer: Springer Nice), (to appear). · Zbl 0991.49500 |
| [6] | Ball, J. M.; Murat, F., Remarks on Chacon’s biting lemma, ESCP Mathl Probl. Nonlinear Mech. Preprint Ser., 14 (1988) · Zbl 0678.46023 |
| [7] | Ball, J. M.; Zhang, K., Lower semicontinuity of multiple integrals and the biting lemma, Proc. R. Soc. Edinb., 114A, 367-379 (1990) · Zbl 0716.49011 |
| [8] | Berliocchi, H.; Lasry, J. M., Integrandes normales et mesures parametres en calcul des variations, Bull. Soc. Math. Fr., 101, 129-184 (1973) · Zbl 0282.49041 |
| [9] | Brooks, J. K.; Chacon, R. V., Continuity and compactness of measures, Adv. Math., 37, 16-26 (1980) · Zbl 0463.28003 |
| [10] | Browder, F. E., Theorems for nonlinear partial differential equations, Proc. Symp. Pure Math., 16, 1-60 (1970) · Zbl 0212.27704 |
| [11] | Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, (Lecture Notes in Mathematics, Vol. 992 (1982), Springer: Springer Berlin) · Zbl 0676.46035 |
| [12] | Eisen, G., A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals, Manuscripta math., 27, 73-79 (1979) · Zbl 0404.28004 |
| [13] | Ekeland, I.; Temam, R., Analyse Convexe et Problèmes Variationnels (1974), Dunod: Dunod Paris · Zbl 0281.49001 |
| [14] | Frehse, J., Existence and perturbation theorems for nonlinear elliptic systems, Preprint No. 576 (1983), SFB 72, Bonn · Zbl 0528.35036 |
| [15] | Landes, R., On the existence of weak solutions of perturbated systems with critical growth, J. reine angew. Math., 393, 21-38 (1989) · Zbl 0664.35027 |
| [16] | Landes, R.; Mustonen, V., On pseudomonotone operators and nonlinear noncoercive variational problems on unbounded domains, Math. Ann., 248, 241-246 (1980) · Zbl 0416.35072 |
| [17] | Neumann, J.von, Functional Operators: Measures and Integrals, Volume I (1950), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0039.28401 |
| [18] | Schonbek, M. E., Convergence of solutions to nonlinear dispersive equations, Communs Partial diff. Eqns, 7, 959-1000 (1982) · Zbl 0496.35058 |
| [19] | Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0207.13501 |
| [20] | Tartar, L.; Knops, R. J., Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, 136-212 (1979) · Zbl 0437.35004 |
| [21] | Zhang, K.-W., On the Dirichlet problem for a class of quasilinear elliptic systems of PDEs in divergence form, (Chern, S. S., Partial Differential Equations, Proc.. Partial Differential Equations, Proc., Lecture Notes in Mathematics, Vol. 1306 (1988), Springer: Springer Berlin), 262-277, Tianjin, 1986 · Zbl 0672.35026 |
| [22] | Zhang, K.-W., Nash point equilibria for variational integrals, I—existence results, Acta Math. Sinica, New ser., 4, 155-176 (1988) · Zbl 0850.49002 |
| [23] | Zhang, K.-W., Biting theorems for Jacobians and their applications, Anal. non lin. Ann. H. Poincaré, 7, 345-365 (1990) · Zbl 0717.49012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
