论文标题
避免使用状态限制的一维非自治功能的Lavrentiev差距
Avoidance of the Lavrentiev gap for one-dimensional non autonomous functionals with state constraints
论文作者
论文摘要
Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional (the "energy"), unnecessarily autonomous, defined on the space of Sobolev functions $W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$).我们认为，在可能满足终点条件或两者兼而有之的功能$ y $中，最小化$ f $的问题。在许多应用中，缺乏规律性或凸度或增长条件并不能确保最小化$ f $的最小化器的存在，那么能够通过一系列Lipschitz函数满足给定边界条件的一系列lipschitz函数近似于$ f $的价值。有时，即使在速度变量中使用某些多项式，强制性和凸的拉格朗日人，从而确保在给定的Sobolev空间中存在最小化器，这是无法实现的：这一事实被称为Lavrentiev现象。本文涉及在\ y Mathcal s \ s \ subset \ subset \ subbb r^n $ in [t，t，t] $的所有$ y（s）\ in \ mathcal s \ subset s \ subset \ subset \ subseT \ subbb r^n $的进一步的状态约束的有效性下避免了lavrentiev现象。 Given $y\in W^{1,p}([t,T];\mathbb R^n)$ with $F(y)<+\infty$ we give a constructive recipe for building a sequence $(y_h)_h$ of Lipschitz reparametrizations of $y$, sharing with $y$ the same boundary condition(s), that converge in energy to $F(y)$.关于以前的有关该主题的文献，我们将（仅）一个终点条件与两者的情况区分开来，将满足足够条件的拉格朗日式扩大，并表明$（y_h）_h $也以$ w^{1，p} $收敛到$ y $。此外，结果也适用于有效领域有限的扩展有价值的Lagrangians。即使Lagrangian是自主的，即即使是$ l（s，y，y'）=λ（y，y'）$的新线索。该论文遵循作者最近的两篇论文\ [23，24]。
Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional (the "energy"), unnecessarily autonomous, defined on the space of Sobolev functions $W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$). We consider the problem of minimizing $F$ among the functions $y$ that possibly satisfy one, or both, end point conditions. In many applications, where the lack of regularity or convexity or growth conditions does not ensure the existence of a minimizer of $F$, it is important to be able to approximate the value of the infimum of $F$ via a sequence of Lipschitz functions satisfying the given boundary conditions. Sometimes, even with some polynomial, coercive and convex Lagrangians in the velocity variable, thus ensuring the existence of a minimizer in the given Sobolev space, this is not achievable: this fact is know as the Lavrentiev phenomenon. The paper deals on the avoidance of the Lavrentiev phenomenon under the validity of a further given state constraint of the form $y(s)\in\mathcal S\subset\mathbb R^n$ for all $s\in [t,T]$. Given $y\in W^{1,p}([t,T];\mathbb R^n)$ with $F(y)<+\infty$ we give a constructive recipe for building a sequence $(y_h)_h$ of Lipschitz reparametrizations of $y$, sharing with $y$ the same boundary condition(s), that converge in energy to $F(y)$. With respect to previous literature on the subject, we distinguish the case of (just) one end point condition from that of both, enlarge the class of Lagrangians that satisfy the sufficient conditions and show that $(y_h)_h$ converge also in $W^{1,p}$ to $y$. Moreover, the results apply also to extended valued Lagrangians whose effective domain is bounded. The results gives new clues even when the Lagrangian is autonomous, i.e., of the form $L(s,y,y')=Λ(y,y')$. The paper follows two recent papers \[23, 24] of the author on the subject.
