具有多值对流和混合边值条件的两相障碍问题
Double phase obstacle problems with multivalued convection and mixed boundary value conditions
摘要：
本文考虑了一个带有双相偏微分算子、障碍效应和多值反应对流项的混合边值问题。在非常一般的假设下，利用多值伪单调算子的满射性定理和Moreau-Yosida的逼近方法，证明了所考虑的混合边值问题的一个存在性定理.然后，我们引入了一族相应于混合边值问题的无约束逼近问题。用$\mathcal S$表示混合边值问题的解集，用$\mathcal S_n$表示逼近问题的解集，建立了如下的收敛关系式\begin{align*} \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*}其中$w$ - $\limsup_{n\to\infty}\mathcal S_n$和$s$ - $\limsup_{n\to\infty}\mathcal S_n$分别表示$\mathcal S_n$的弱Kuratowski上限和强Kuratowski上极限.
英文摘要：
In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $\mathcal S$ the solution set of the mixed boundary value problem and by $\mathcal S_n$ the solution sets of the approximating problems, we establish the following convergence relation \begin{align*} \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} where $w$-$\limsup_{n\to\infty}\mathcal S_n$ and $s$-$\limsup_{n\to\infty}\mathcal S_n$ stand for the weak and the strong Kuratowski upper limit of $\mathcal S_n$, respectively.
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