<?xml version="1.0" encoding="utf-8" standalone="yes"?>
<item>
  <id>04198613</id>
  <dt>a</dt>
  <an>04198613</an>
  <augroup>
    <au>Filippov, V.M.</au>
  </augroup>
  <ti>On the variational method for ultraparabolic equations.</ti>
  <so>Analysis of information-computational systems, Collect. Sci. Works, Moskva, 107-111 (1986).</so>
  <py>1986</py>
  <pu></pu>
  <lagroup>
    <la>RU</la>
  </lagroup>
  <ccgroup>
  </ccgroup>
  <utgroup>
    <ut>existence</ut>
    <ut>uniqueness</ut>
    <ut>minimum element</ut>
  </utgroup>
  <cigroup>
    <ci>Zbl 0696.00028</ci>
  </cigroup>
  <ligroup>
  </ligroup>
  <abgroup>
    <ab>[For the entire collection see Zbl 0696.00028.] Let the bounded domain $\Omega =\{(x,t)=(x,t\sb 1,...,t\sb N):$ $a<x<b,0<t\sb i<T\sb i$, $1\le i\le N\}$ be given, and $k(x,t)\in C\sp 1(\Omega)\cap C({\bar \Omega})$, k(x,t)$\ge \alpha$ for all (x,t)$\in \Omega$, where $\alpha$ is a positive constant, $a\sb i(x,t)\in C({\bar \Omega})$, $a\sb i(x,t)\vert\sb{t\sb i=0}=a\sb i(x,t)\vert\sb{t\sb i=T\sb i}$, $\vert a\sb i(x,t)\vert =0$, (x,t)$\in \Omega$, $1\le i\le N$. Let $\Gamma\sb{\beta}=\{(x,t):$ $x=\beta$, $0\le t\sb i\le T\sb i$, $1\le i\le N\}$, $\beta =a,b$. Consider the following boundary value problem: (1) $u(x,t)\vert\sb{t\sb i=0}=u(x,t)\vert\sb{t\sb i=T\sb i}$ (1$\le i\le N$, $a\le x\le b);$ (2) $ tial u/ tial n=\psi (t)$, $(x,t)\in \Gamma\sb a$, where n is the outer normal of $\Omega$ ; (3) $u(x,t)=0$, $(x,t)\in \Gamma\sb b.$ Consider the ultraparabolic equation (4) ${\cal L}\sb pu\equiv \sum\sp{N}\sb{i=1}a\sb i(x,t) tial u/ tial t\sb i-(k(x,t)u\sb x)=g(x,t)$, (x,t)$\in \Omega.$ The author solves the boundary value problem (1)-(4) by a variational method as follows. Denote $\overset \circ \to C\sp{2,1}({\bar\Omega},\Gamma\sb b,T)=\{u(x,t)\in C\sp{2,1}\sb{x,t}({\bar \Omega})$ satisfying (1) and (3)$\}$. Let ${\cal L}u=\{{\cal L}\sb pu$; $k tial u/ tial n\}$ and $Bu=\{B\sb pu;B\sb pu\}$ where $B\sb pu=u- \sum\sp{N}\sb{1}\int\sp{b}\sb{x}(d\theta /k(\theta,t))\int\sp{\theta}\sb{a}a\sb i(\xi,t)( tial u/ tial t\sb i)(\xi,t)d\xi.$ Under the restriction $\sum\sp{N}\sb{i=1} tial a\sb i/ tial t\sb i\le 0$, (x,t)$\in \Omega$, finding the generalized solution of (1)-(4) is equivalent to finding a $u\in \overset \circ \to C\sp{2,1}({\bar \Omega},\Gamma\sb b,T)$ giving the minimum of the functional $$ D(u)=\int\sb{\Omega}{\cal L}\sb puB\sb pudx dt+\int\sb{\Gamma\sb a}k( tial u/ tial n)B\sb pd\Gamma\sb a- 2(f,Bu), $$ for all $u\in D(B)$. The author shows the existence and uniqueness of the minimum element for D(u).</ab>
    <rv></rv>
  </abgroup>
</item>