论文标题
可变指数双相问题的多价变分不平等问题：比较和极端结果
Multi-valued variational inequalities for variable exponent double phase problems: comparison and extremality results
论文作者
论文摘要
我们证明了形式的有限域$ω$的多价变量不等式的存在和比较结果，\ begin {equation*} u \ in K \，：\，0 \，在au++\ partial i_k（partial i_k（u）+\ \ \ \ \ \ \ \ \ \ \ {f}（f}（f}（u）+\ m natrycal+\ m natrcal in au+\ partial i_k（u）中， } w^{1，\ Mathcal {h}}（ω）^*，\ end {equation*}其中$ a \ colon w^{1，\ nathcal {h}}}（ω）\ to w^{1，w^{1，\ mathcal {h}}}} au：= - \ text {div} \ left（| \ nabla u |^{p（x）-2} \ nabla u+++++++μ（x）| \ nabla u |^{q（x）-2} \ nabla u \ right）具有可变指数的双相运算符和$ w^{1，\ Mathcal {h}}（ω）$是关联的Musielak-Orlicz sobolev Space。首先，在某些较弱的顽强条件下证明了存在结果。我们的主要重点旨在治疗在强制性失败时考虑的问题。为此，我们建立了基于适当定义的子和超术的空间中的多价变化不平等的子谋杀方法$ w^{1，\ Mathcal {H}}}（ω）$，从而产生了在子次数的有序间隔内解决方案的存在。此外，如果封闭的凸子集$ k $ $ w^{1，\ Mathcal {h}}}（ω）$满足晶格条件，则将显示极端解决方案的存在。作为亚启动方法的应用，我们能够表明，与领先的双相操作员一起使用一类普遍的变异 - 杀菌性不平等现象，作为此处考虑的多价值变异不平等的特殊情况。基于固定点参数，我们还研究了相应的Nemytskij操作员$ \ MATHCAL {F}，\ MATHCAL {F}_γ$不必连续的情况。最后，我们给出了与上述问题相关的子和超扫描构建的非平凡例子。
We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $Ω$ of the form \begin{equation*} u\in K\,:\, 0 \in Au+\partial I_K(u)+\mathcal{F}(u)+\mathcal{F}_Γ(u)\quad\text{in }W^{1,\mathcal{H}}(Ω)^*, \end{equation*} where $A\colon W^{1, \mathcal{H}}(Ω) \to W^{1, \mathcal{H}}(Ω)^*$ given by \begin{equation*} Au:=-\text{div}\left(|\nabla u|^{p(x)-2} \nabla u+ μ(x) |\nabla u|^{q(x)-2} \nabla u\right) \end{equation*} for $u \in W^{1, \mathcal{H}}(Ω)$, is the double phase operator with variable exponents and $W^{1, \mathcal{H}}(Ω)$ is the associated Musielak-Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub-supersolution for the multi-valued variational inequality in the space $W^{1, \mathcal{H}}(Ω)$ based on appropriately defined sub- and supersolutions, which yields the existence of solutions within an ordered interval of sub-supersolution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset $K$ of $W^{1, \mathcal{H}}(Ω)$ satisfies a lattice condition. As an application of the sub-supersolution method we are able to show that a class of generalized variational-hemivariational inequalities with a leading double phase operator are included as a special case of the multi-valued variational inequality considered here. Based on a fixed point argument, we also study the case when the corresponding Nemytskij operators $\mathcal{F}, \mathcal{F}_Γ$ need not be continuous. At the end, we give a nontrivial example of the construction of sub- and supersolutions related to the problem above.
