Bounded Positive Critical Points of Some Multiple Integrals of the Calculus of Variations
Lucio Boccardo, Benedetta Pellacci · Birkhäuser Basel eBooks · 2003
Let 52 be a bounded domain in E N (N > 2) and a: ¨ª x JR ¡ª IR be a bounded smooth real function. We recall that simple functionals as are differentiable only along directions of W 0 1 ’ 2 (Q) n L¡ã¡ã (e). The directional derivative is given by where the lower order term a s (x , u) I VuI2, in general, belongs to L 1 (Q) and not to W 1 2 (f). Existence results of critical points of such functionals have been proved in [2], [3], [4], [5], [7], [8], [9], [10], [11]. In [7], [9], [10] some multiplicity result are proved by using a “weak” notion of derivative for continuous functions in complete metric spaces. In [2], [3], [4], some existence results are proved for bounded and unbounded coefficients a(x , s). Here we will present some existence results for this problem, proved in [2], [5] and [11]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.