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In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to functions in . It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...
We study the problem of minimizing over the functions that assume given boundary values on . The lagrangian and the domain are assumed convex. A new type of hypothesis on the boundary function is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...
We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models iswith a convex function with general growth (also exponential behaviour is allowed).
We prove Lipschitz continuity for local
minimizers of integral functionals of the Calculus of Variations
in the vectorial case, where the energy density depends explicitly
on the space variables and has general growth with respect to the
gradient. One of the models is
with h a convex function with general growth (also exponential behaviour
is allowed).
The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincare-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling...
Continuing the previous Part I, the degenerate first order variational integrals depending on two functions of one independent variable are investigated.
We show that local minimizers of functionals of the form
, ,
are locally Lipschitz continuous provided f is a convex function with growth
satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous
in u. As a consequence of this, we obtain an existence result for a related nonconvex
functional.
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