Equations de type implicite avec contraintes

Chiara Tanteri
2000
EPFL thesis

In this thesis we will treat the Dirichlet problem for systems of implicit equations, i.e. where Ω ⊂ Rn is an open set, u : Ω → Rm, Fi : Ω × Rm × Rm×n → R, m,n ≥ 1, are continuous functions and φ, the boundary datum, is given. At first we will be interested in the study of problems of the type (1) under constraints. We will show theorems of existence of Lipschitz solutions using an approach based on Baire category theorem. As corollaries of our abstract results, we will give two theorems related to the constraints det Du > 0 and det Du = 1, the constraints most closely related to the applications. Indeed, the constraints det Du > 0 and det Du = 1 come from nonlinear elasticity and represent respectively the conditions of non interpenetration of matter and incompressibility. In the second part of this study, we will focus on the applications. We will treat many examples such as the case of singular values, potential wells under the incompressibility constraint (i.e. det Du = l), the problem of confocal ellipses, the problem of nematic elastomers, particularly related to the fields of nonlinear elasticity, the microstructure of the crystals and the optimal design, as well as the complex eikonal equation (application related to geometric optics). From the mathematical point of view, we will give sufficient conditions of solvability of the system (1). These conditions consist in characterizing the different convex hulls. Indeed, the possibility of representing these sets, in algebraic terms, gives one of the conditions which the boundary datum must satisfy so that a problem of the type (1) admits a solution. Finally we will extend to polyconvex sets properties such as the gauge, the characterization of the extreme points and the Choquet function, all of which are well-known tools within the framework of classical convex analysis.

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Related concepts (6)

Theorem

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic.

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).

Convex hull

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact.