Implicit Partial Differential Equations

Abstract

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.

Viscosity solutions, almost everywhere solutions and explicit formulas

“…The function L being convex we have that D + L (x) (the superdifferential of L at x; see [1] and [6] for the precise definition of this set) is either empty or reduced to {DL (x)}, i.e. x is a point of differentiability of L and we know by Theorem 1 that at such points DL (x) ∈ E. We therefore have that…”

“…We have the following theorem that is inspired by Cardaliaguet-DacorognaGangbo-Georgy [3] (see also [6]). …”

“…We recall the following two facts (the first one is just the HopfLax formula and the second one is Lemma 2.9 in [3] or Lemma 4.17 in [6]). We also use the standard notation D + u (x), respectively D − u (x), for the superdifferential, respectively the subdifferential, of u at x (see [6] for more details).…”

“…x ∈ Ω, u (x) = 0, x∈ ∂Ω , has recently been extensively studied in the book [6] by the authors. Here F : R n → R is a continuous function and we look for a Lipschitz-continuous solution u : Ω ⊂ R n → R. A wide literature on this subject can be found in [6], not only for scalar problems such as this one, but also for vector-valued solutions of first order systems related to maps u : Ω ⊂ R n → R m and F : R m×n → R N , for some m, N ≥ 1.…”

“…It has been studied by many authors starting with Hopf, Lax, Kruzkov and Crandall-Lions; see for example [1] or [6] for more historical comments. One of the earliest and still one of the most complete monographs on the subject is [10] by P.L.…”