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Abstract. We study a Dirichlet problem associated to some nonlinear partial di¤erential equations under additional constraints that are relevant in non linear elasticity. We also give several examples related to the complex eikonal equation, optimal design, potential wells or nematic elastomers.

where D is a (bounded) open set of R n, F: R m• and ~E W 1'~ (D; Rm). We emphasize that u: DcRn--*R m, with m, n~>l, is a vector valued function if m> 1 (otherwise, if m--l, we say that u is a scalar function). As usual Du denotes the gradient of u. This problem (1.1) has been intensively studied, essentially in the scalar case in many relevant articles such as Lax [28], Douglis [23], Kru2kov [27], Crandall-Lions [16], Crandall-Evans-Lions [14], Capuzzo Dolcetta-Evans [8], Capuzzo Dolcetta-Lions [9], Crandall-Ishii-Lions [15]. For a more complete bibliography we refer to the main recent monographs of Benton [7], Lions [29], Fleming-Soner [25], Barles [6] and Bardi-Capuzzo Dolcetta [5]. B. DACOROGNA AND P. MARCELLINI Our motivation to study this equation, besides its intrinsic interest, comes from the calculus of variations. In this context first order partial differential equations have been intensively used, cf. for example the monographs of Carath@odory [10] and Rund [36] (for more recent developments on the vectorial case, see [19]). In this paper we propose some new hypotheses on the function F in (1.1) that allow us to treat systems of equations as well as vectorial problems (cfi examples below). The general existence result (Theorem 2.1) can be applied to the following examples, thatfor the sake of simplicity we state under the additional assumption that the boundary datum qa is of class Cl(~t;Rm).

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