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Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. (English) Zbl 0602.35030
We consider the following problem: \[ (0_{\epsilon})\quad - a_{ij}(x,x/\epsilon)\partial^ 2u^{\epsilon}/\partial x_ i \partial x_ j-(1/\epsilon)b_ i(x,x/\epsilon)\partial u^{\epsilon}/\partial x_ i=H(x,x/\epsilon),u^{\epsilon},Du^{\epsilon}) \] in \({\mathcal O}\), \(u_{\epsilon}=0\) on \(\Gamma\).
Here the coefficients \(a_{ij}\) and \(b_ i\) are smooth, periodic with respect to the second variable, and the matrix \((a_{ij})_{ij}\) is uniformly elliptic. The Hamiltonian H is locally lipschitz continuous with respect to u and Du, and has quadratic growth in Du. The Hamilton- Jacobi-Bellman equations of some stochastic control problems are of this type.
Our aim is to pass to the limit in \((0_{\epsilon})\) as \(\epsilon\) tends to zero. We assume the coefficients \(b_ i\) to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive \(L^{\infty}\), \(H^ 1_ 0\) and \(W^{1,p_ 0}(p_ 0>2)\) estimates for the solutions of \((0_{\epsilon})\). We also prove the following corrector’s result: \[ Du^{\epsilon}-Du+D\chi (x,x/\epsilon)Du\to 0\quad in\quad (L^ 2({\mathcal O}))^ N\quad strongly. \] This allows one to pass to the limit in \((0_{\epsilon})\) and to obtain: \[ (0_ 0)\quad -q_{ij}(x) \partial^ 2u/\partial x_ i \partial x_ j-r_ i(x) \partial u/\partial x_ i={\mathcal H}(x,u,Du)\quad in\quad {\mathcal O},\quad u=0\quad on\quad \Gamma. \] This problem is of the same type as the initial one. When \((0_{\epsilon})\) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then \((0_ 0)\) is also a Hamilton-Jacobi-Bellman equation but corresponding to modified set of controls.

35J60 Nonlinear elliptic equations
49L99 Hamilton-Jacobi theories
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Amann, Indiana Univ. Math. J. 27 pp 779– (1978)
[2] Stochastic Control by Functional Analysis Methods, North Holland, 1982.
[3] Homogenization theory. Atti del S.A.F.A. III, Bari 27/9–5/10, 1978, Conferenze del Seminario Matematico dell’Universita di Bari 158, 1979.
[4] Bensoussan, J. für die Reine und Angewandte Mathematik 350 pp 23– (1984)
[5] , and , Asymptotic Analysis for Periodic Structures, North Holland, 1978.
[6] and , Homogénéisation de problémes quasi-linéaires, Atti del Convegno Studio di Problemi-limite dell’Analisi Funzionale, Bressanone, 7–9 Settembre, 1981, Pitagora Editrice, 1982, pp. 13–51.
[7] , and , Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique, Non-linear Partial Differential Equations and their Applications, Collège de France Seminar, vol. IV, ed. by and , Research Notes in Math. 84, Pitman, 1983, pp. 19–73.
[8] Boccardo, Ann. Sc. Norm. Sup. Pisa 11 pp 213– (1984)
[9] De Giorgi, Boll. Un. Mat. Ital. 8 pp 391– (1973)
[10] Stochastic Processes, Wiley, New York, 1967.
[11] and , Analyse Convexe et Problèmes Variationnels, Dunod, Paris, 1974. · Zbl 0281.49001
[12] Partial Differential Equations of Parabolic Type, Prentice Hall, 1964.
[13] Gehring, Acta. Math. 130 pp 265– (1973)
[14] Giaquinta, J. für die Reine und Angewandte Mathematik 311/312 pp 145– (1979)
[15] and , Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977.
[16] and , Equations aux Dérivées Partielles de Type Elliptique, Dunod, Paris, 1968.
[17] Meyers, Ann. Sc. Norm. Sup. Pisa 17 pp 189– (1963)
[18] Spagnolo, Ann. Sc. Norm. Sup. Pisa 22 pp 571– (1968)
[19] Cours Peccot, Collège de France, 1977; partially written in Murat, F., H-convergence, Séminaire d’Analyse Fonctionnelle et Numerique de l’Université d’Alger, 1977 –1978.
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