Abstract We consider the classical functional of the Calculus of Variations of the form I(u)=∫ΩF(x,u(x),∇u(x))dx, $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + W01,p $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

中文翻译：

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一类凸非自治拉格朗日函数的 Lavrentiev 现象不发生

摘要 我们考虑 I(u)=∫ΩF(x,u(x),∇u(x))dx, $$\begin{array}{} \displaystyle I( u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ 其中 Ω 是ℝn 和 F : Ω × ℝ × ℝn → ℝ 是 Carathéodory 凸函数；容许函数 u 与 ∂Ω 上的指定 Lipschitz 函数 ϕ 一致。我们制定了一些条件，其中 ϕ + W01,p $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) 且 I(u) < +∞可以通过一系列与 ∂Ω 上的 ϕ 重合的平滑函数在 W1,p 范数和能量中近似。作为一个特殊情况，我们得到当 F(x, u, ξ) = f(x, u) + h(x, ξ) 是凸的并且 x ↦ F(x, 0, 0) 是足够光滑。