论文标题
关于没有增长假设的Euler-Lagrange系统的有效性
On the validity of the Euler-Lagrange system without growth assumptions
论文作者
论文摘要
凸$ \ mathscr f（v）= \int_Ωf（\ nabla^k v（x））\ mathrm d x $在sobolev mappings $ v \ in \ mathrm w^^{ dirichlet类$ \ mathrm w^{k，1} _ {g}（ω，\ mathbb r^n）的子集，$的特征是$ \ mathscr f $的Euler-Lagrange不平等的能量解决方案。 We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity.在不受约束的情况下，$ k = \ mathrm w^{k，1} _ {g}（ω，\ \ mathbb r^n）$，如果intemperand $ f $是凸面，实现的，并满足demi-coercerivity条件，那么$$int_Ω\ \ \ \！ f^{\ prime}（\ nabla^{k} u）\ cdot \ nabla^{k} ϕ \，\ mathrm d x = 0 $$保留所有$ ϕ \ in \ in \ mathrm w_ {0}矢量测量的绝对连续部分$ d^{k} u $。
The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_ΩF(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(Ω, \mathbb R^N )\cap K$, where $K$ is a closed convex subset of the Dirichlet class $\mathrm W^{k,1}_{g}(Ω, \mathbb R^N ),$ are characterised as the energy solutions to the Euler-Lagrange inequality for $\mathscr F$. We assume that the essentially smooth integrand $F\colon \mathbb R^{N} \otimes \odot^{k}\mathbb R^{n} \to \mathbb R\cup\{+\infty\}$ is convex, lower semi-continuous, proper and at least super-linear at infinity. In the unconstrained case $K=\mathrm W^{k,1}_{g}(Ω, \mathbb R^N )$, if the integrand $F$ is convex, real-valued, and satisfies a demi-coercivity condition, then $$ \int_Ω \! F^{\prime}(\nabla^{k} u) \cdot \nabla^{k}ϕ\, \mathrm d x =0 $$ holds for all $ϕ\in \mathrm W_{0}^{k}( Ω, \mathbb R^{N})$, where $\nabla^{k} u$ is the absolutely continuous part of the vector measure $D^{k}u$.
