论文标题
一维非自主函数的间隙不存在
Non-occurrence of gap for one-dimensional non autonomous functionals
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, unnecessarily autonomous, defined on the space $ W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$) of Sobolev functions, possibly with prescribed one or two end point conditions.重要的是，尤其是对于应用程序而言，能够沿一系列满足相同边界条件（S）的Lipschitz函数将$ f $的最小值与$ f $的值近似。有时，这是不可能的，即，所谓的Lavrentiev现象发生。像Manià的Lagrangian $ l（s，y，y'）=（y^3-s）^2（y'）^6 $和边界数据$ y（0）= 0，y（1）= 1 $一样，就是这种无辜的情况。然而，在这种情况下，仅在终点条件$ y（1）= 1 $的情况下，差距不会发生。本文着重于避免仅一个或两个终点条件的问题所需的不同条件集。在对可能延长的拉格朗日（Lagrangian）的最小假设下，我们确保只有一个终点条件的Lavrentiev现象的不存在。我们介绍了一个额外的假设，当拉格朗日在有限集中界定时满足，以确保两个终点条件的有效性$ y_h（t）= y（t），y_h（t）= y（t）$;即使在自主情况下，结果也给了一些新的光。
Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional, unnecessarily autonomous, defined on the space $ W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$) of Sobolev functions, possibly with prescribed one or two end point conditions. It is important, especially for the applications, to be able to approximate the infimum of $F$ with the values of $F$ along a sequence of Lipschitz functions satisfying the same boundary condition(s). Sometimes this is not possible, i.e., the so called Lavrentiev phenomenon occurs. This is the case of the innocent like Manià's Lagrangian $L(s,y,y')=(y^3-s)^2(y')^6$ and boundary data $y(0)=0, y(1)=1$; nevertheless in this situation the gap does not occur with just the end point condition $y(1)=1$. The paper focuses about the different set of conditions needed to avoid the gap for problems with just one or with both end point conditions. Under minimal assumptions on the, possibly extended valued, Lagrangian we ensure the non-occurrence of the Lavrentiev phenomenon with just one end point condition. We introduce an additional hypothesis, satisfied when the Lagrangian is bounded on bounded sets, in order to ensure the validity of both end point conditions $y_h(t)=y(t), y_h(T)=y(T)$; the result gives some new light even in the autonomous case.
