{"id":4100,"date":"2013-11-12T23:33:55","date_gmt":"2013-11-12T15:33:55","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=4100"},"modified":"2013-11-12T23:33:55","modified_gmt":"2013-11-12T15:33:55","slug":"weak-convergence-in-sobolev-spaces","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=4100","title":{"rendered":"Weak Convergence in Sobolev Spaces"},"content":{"rendered":"<p>Suppose $\\Omega\\subset\\R^n$ and donote $W^{1,p}:=W^{1,p}(\\Omega)$ be the sobolev space for some $1&lt; p&lt; +\\infty$. Recall that $f_i\\in W^{1,p}$ convergent weakly to $f\\in W^{1,p}$, if for any $\\phi$ in the dual space of $W^{1,p}$, we have $\\inner{f_i,\\phi}\\to\\inner{f,\\phi}$, denote as $f_i\\weakto f$. This is distinguished by strongly convergence, as we use the dual normal instead of $W^{1,p}$ normal.<br \/>\n\\begin{prop}<br \/>\nIf $f_i\\weakto f$ in $W^{1,p}$, then $f_i\\to f$ in $L^p$.<br \/>\n\\end{prop}<br \/>\n<!--more--><br \/>\n\\begin{proof}<br \/>\nSince $W^{1,p}$ is a Banach space, and $f_i\\weakto f$, then $f_i$ is uniformly bounded in $W^{1,p}$ by <strong>Banach-Steinhaus Theorem<\/strong>. Since $W^{1,p}$ is compactly embedding into $L^p$, we have, by passing to subsequence, $f_i\\to g$ in $L^p$ for some $g\\in L^p$.<br \/>\nFor any $\\phi\\in C_0^\\infty$, define a $T_\\phi$ as<br \/>\n$$<br \/>\nT_\\phi(f):=\\int_\\Omega f\\phi,\\quad\\forall f\\in W^{1,p},<br \/>\n$$<br \/>\nthen $T_\\phi\\in (W^{1,p})&#8217;$, i.e., in the dual space of $W^{1,p}$. Thus, by weakly convergence, we have<br \/>\n$$<br \/>\nT_\\phi(f_i)\\to T_\\phi(f)\\Longleftrightarrow \\int_\\Omega f_i\\phi\\to\\int_\\Omega f\\phi.<br \/>\n$$<br \/>\nOn the other hand, $f_i\\to g$ strongly in $L^p$, thus by Holder inequality, we have<br \/>\n$$<br \/>\n\\int_\\Omega f_i\\phi\\to\\int_\\Omega g\\phi.<br \/>\n$$<br \/>\nIn conclusion, we have<br \/>\n$$<br \/>\n\\int_\\Omega f\\phi=\\int_\\Omega g\\phi,<br \/>\n$$<br \/>\nand we finish the argument by the density of $C_0^\\infty$ in $L^p$.\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Suppose $\\Omega\\subset\\R^n$ and donote $ &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=4100\"> <span class=\"screen-reader-text\">Weak Convergence in Sobolev Spaces<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[626,669],"class_list":["post-4100","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-sobolev-spaces","tag-weakly-convergence"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4100"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4100\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4100"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4100"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}