SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1489-1525, January 2020.
In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions---both weak and variational solutions---to the associated $L^2$-gradient flow. The initial values are merely $L^2$ functions with possibly infinite energy. In this context, the notion of integral convexity, i.e., the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and nonconvex integrands.

中文翻译：

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整体凸性和抛物线系统

SIAM数学分析杂志，第52卷，第2期，第1489-1525页，2020
年1月。在这项工作中，我们给出了变量微积分的最优（即必要和充分）条件，以保证解的存在-无论是弱解还是变分解-到相关的$ L ^ 2 $梯度流。初始值只是具有无限能量的$ L ^ 2 $函数。在这种情况下，积分凸度的概念，即变分积分而不是被积数的凸度，起着至关重要的作用。令人惊讶的是，这种类型的凸度比被积物的凸度弱。我们通过某些拟凸和非凸被积数证明了这一点。