关于趋于无穷域上的某些变分问题
On some Variational Problems set on domains tending to infinity
摘要：
设$Ω_\ell = \ellω_1 \times ω_2$，其中$ω_1 \subset \R^p$和$ω_2 \subset \R^{n-p}$被假定为开有界的。我们考虑以下最小化问题：$$E_{Ω_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(Ω_\ell)}E_{Ω_\ell}(u)$$其中$E_{Ω_\ell}(u) = \int_{Ω_\ell}F(\grad u)-fu$，$F$是凸函数，$f\in L^{q'}(ω_2)$ .我们感兴趣的是研究解$u_\ell$在$\ell$趋于无穷大时的渐近行为。
英文摘要：
Let $Ω_\ell = \ellω_1 \times ω_2$ where $ω_1 \subset \R^p$ and $ω_2 \subset \R^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{Ω_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(Ω_\ell)}E_{Ω_\ell}(u)$$ where $E_{Ω_\ell}(u) = \int_{Ω_\ell}F(\grad u)-fu$, $F$ is a convex function and $f\in L^{q'}(ω_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.
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