Convergence analysis for double phase obstacle problems with multivalued convection term
Shengda Zeng, Yunru Bai, Leszek Gasiński, Patrick Winkert
Abstract
Abstract In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢 n the solution sets of approximating problems, we prove the following convergence relation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mi mathvariant="normal">∅</m:mi> <m:mo>≠</m:mo> <m:mi>w</m:mi> <m:mtext>-</m:mtext> <m:munder> <m:mo>lim sup</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:msub> <m:mrow class="MJX-TeXAtom-ORD"> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">S</m:mi> </m:mrow> </m:mrow> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mi>s</m:mi> <m:mtext>-</m:mtext> <m:munder> <m:mo>lim sup</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:msub> <m:mrow class="MJX-TeXAtom-ORD"> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">S</m:mi> </m:mrow> </m:mrow> <m:mi>n</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">S</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{array}$$ where w -lim sup n →∞ 𝓢 n and s -lim sup n →∞ 𝓢 n denote the weak and the strong Kuratowski upper limit of 𝓢 n , respectively.