We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property.
The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group.
Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler-Lagrange system of PDE.
A rigidity result for a global vectorial free boundary problem with a linear datum
Dipartimento di Matematica Universita di Pisa（比萨大学数学系）；Centro di Ricerca Matematica Ennio De Giorgi Scuola Normale Superiore di Pisa（比萨高等师范学校恩尼奥·德乔治数学研究中心）
On the blow-up of the vectorial Bernoulli free boundary problem
Dipartimento di Matematica Universita di Pisa（比萨大学数学系）；Centro di Ricerca Matematica Ennio De Giorgi Scuola Normale Superiore di Pisa（比萨高等师范学院恩尼奥·德乔治数学研究中心）
On a Multiphase Vectorial Bernoulli Free Boundary Problem
Dipartimento di Matematica Universita di Pisa（比萨大学数学系）；Centro di Ricerca Matematica Ennio De Giorgi Scuola Normale Superiore di Pisa（比萨高等师范学校恩尼奥·德乔治数学研究中心）
Autonomous Integral Functionals with Discontinuous Nonconvex Integrands: Lipschitz Regularity of Minimizers, DuBois-Reymond Necessary Conditions, and Hamilton-Jacobi Equations
Swarm gradient dynamics for global optimization: the density case
使用联合几何和随机重新制定的非凸问题，并利用Monge-Kantorovich梯度系统制定消失的力量，我们正式扩展的模拟退火方法的广泛的一类全球最优化方法。由于类梯度策略和粒子相互作用的内置组合，我们称之为群梯度动力学。正如Holley-Kusuoka-Stroock的原始论文一样，确保收敛到全局最小值的时间表存在的关键是一个函数不等式。我们的中心理论贡献之一是证明这样一个不等式的一维紧流形。我们推测这个不等式在更广泛的背景下是成立的。我们还描述了一个通用的方法，允许全局优化和证明的关键作用的功能不平等{à} LA ojasiewicz。
