论文标题
有界线性函数扰动的二次函数的伽马连接
Gamma-convergence of quadratic functionals perturbed by bounded linear functionals
论文作者
论文摘要
给定一个有界的开放集$ω\ subset \ mathbb {r}^n $，我们研究了sobolev space上二次函数的序列$ h^1_0（ω）$，受到有界线性函数序列的扰动。我们证明，在$ H^1_0（ω）$的弱拓扑中，他们的$γ$限制始终可以写入二次功能，线性功能和非阳性常数的总和。 $ g $ - 和$ h $ -convergence的经典理论完全表征了$γ$限制的二次和线性部分，并表明它们的系数不取决于$ω$。相反，该常数取决于$ω$，并用$-ν（ω）$表示，在研究解决方案能量的极限行为中起着重要作用。本文的主要结果是，传递到子序列，我们可以证明$ν$与非负ra量相吻合，这是在足够大的有界开放式$ω$的大量集合上。此外，我们展示了一个示例，该示例表明，每个有限的开放集都无法获得先前的结果。该示例的具体形式表明，在$γ$ -Convergence中的定位方法的紧凑定理无法轻易改善。
Given a bounded open set $Ω\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(Ω)$, perturbed by sequences of bounded linear functionals. We prove that their $Γ$-limits, in the weak topology of $H^1_0(Ω)$, can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of $G$- and $H$-convergence completely characterises the quadratic and linear parts of the $Γ$-limit and shows that their coefficients do not depend on $Ω$. The constant, which instead depends on $Ω$ and will be denoted by $-ν(Ω)$, plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that $ν$ coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets $Ω$. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in $Γ$-convergence cannot be easily improved.
