### split into another section

parent 834eb326
Pipeline #8032 passed with stages
in 41 seconds
 ... ... @@ -313,7 +313,7 @@ Using distribution theory\footnote{see \url{https://en.wikipedia.org/wiki/Distri To make sense of the expression $D u$ here, one has to use distribution theory, i.e. the \Index{distribution} $D u$ is required to be an actual Element of $L^2(M,E)$. \end{remark} Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\ fullfil $\overline{D}=D^*$. Furthermore a symmetric differential operator may be \Index{essentially selfadjoint} (or just \Index{selfadjoint}), i.e.\ fullfil $\overline{D}=D^*$. Note that the closure $\overline{D}$ is an extension of $D$ in general and that $D^*$ is an extension of $\overline{D}$ in the symmetric case. As the following example shows, all three operators may be different: ... ... @@ -327,7 +327,18 @@ As the following example shows, all three operators may be different: \end{example} This example already suggests, that non-selfadjointness is in some way connected to \enquote{boundary conditions} on a non-compact manifold $M$. The same phenomenon shows up in the following statement: We investigate this phenomenon in the next section and conclude the present section with the following statement about elliptic operators. \begin{lemma}\label{lem:adjoint_elliptic} If $D$ is a symmetric, essentially selfadjoint elliptic operator, then its adjoint is elliptic as well. \end{lemma} \begin{proof} \todo{add proof} \end{proof} % section hilbert_space_techniques_and_the_formal_adjoint (end) \section{Selfadjointness of Differential Operators} % (fold) \label{sec:self_adjointness_of_differential_operators} \begin{lemma}\label{lem:symmetric_ess_sa} Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$ \emph{compactly supported}. ... ... @@ -367,15 +378,9 @@ We need to look at multiplication operators. Then $M$ is called \Index{complete} for $D$ if there is a smooth proper function $g \colon M \to \mathbb{R}$ such that $\benbrace*{D,\rho(g)}$ is a bounded Hilbert space operator. \end{definition} \todo[inline]{include wave operator stuff as well?} \begin{lemma}\label{lem:adjoint_elliptic} If $D$ is a symmetric elliptic operator, then its adjoint is elliptic as well. \end{lemma} \begin{proof} \todo{add proof} \end{proof} % section hilbert_space_techniques_and_the_formal_adjoint (end) \todo[inline]{include wave operator stuff as well?} % section self_adjointness_of_differential_operators (end) \section{Analysis of Elliptic Differential Operators} % (fold) \label{sec:analysis_of_elliptic_differential_operators} ... ...
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
{"ft_lang_label":"__label__en","ft_lang_prob":0.7288522,"math_prob":0.98602706,"size":2650,"snap":"2020-24-2020-29","text_gpt3_token_len":744,"char_repetition_ratio":0.13945578,"word_repetition_ratio":0.17197452,"special_character_ratio":0.2671698,"punctuation_ratio":0.16331096,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982859,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-06T01:43:40Z\",\"WARC-Record-ID\":\"<urn:uuid:384a3852-85fc-4515-82a0-a32bb450210e>\",\"Content-Length\":\"111101\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ce2d7111-e53b-4220-97d8-6de5be2410cb>\",\"WARC-Concurrent-To\":\"<urn:uuid:f2179d17-c094-4809-bc0b-a5f9dec3786c>\",\"WARC-IP-Address\":\"128.176.11.34\",\"WARC-Target-URI\":\"https://zivgitlab.uni-muenster.de/j_bant01/differential-operators/-/commit/bd382af55516a0afd44f75fcd250bd9825fc897e\",\"WARC-Payload-Digest\":\"sha1:HI7RUPWXLS2SFL2GEAPTQPKCJR2ZKBDV\",\"WARC-Block-Digest\":\"sha1:KPLAXLZNGW2AYX5XYRTZSMK7CSG5RCUK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590348509264.96_warc_CC-MAIN-20200606000537-20200606030537-00300.warc.gz\"}"}