论文标题
具有一维标量自主函数的差距的出现
Occurrence of gap for one-dimensional scalar autonomous functionals with one end point condition
论文作者
论文摘要
令$ l：\ Mathbb r \ times \ Mathbb r \ to [0， +\ infty [\，\ cup \ { +\ infty \} $是Borel功能。我们考虑问题\ begin {equination} \ tag {p} \ min f（y）= \ int_0^1l（y（t），y'（t））\，dt：y（0）= 0，\，y \ in w^{1,1}（1,1}（1,1}（[0,1]，[0,1]，\ mathbb r r）$ nequian agr。为此发生Lavrentiev现象。我们陈述了一个条件，仅涉及两个函数图上$ l $的行为，从而确保了现象的不存在。我们的标准大大削弱了众所周知的条件，即$ l $在有限的集合上有限。
Let $L:\mathbb R\times \mathbb R\to [0, +\infty[\,\cup\{+\infty\}$ be a Borel function. We consider the problem \begin{equation}\tag{P}\min F(y)=\int_0^1L(y(t), y'(t))\,dt: y(0)=0,\, y\in W^{1,1}([0,1],\mathbb R).\end{equation} We give an example of a real valued Lagrangian $L$ for which the Lavrentiev phenomenon occurs. We state a condition, involving only the behavior of $L$ on the graph of two functions, that ensures the non-occurrence of the phenomenon. Our criterium weakens substantially the well-known condition, that $L$ is bounded on bounded sets.
