| Research Abstract |
This research is concerned with equation :
(1) uィイD2tィエD2-Δu+Fu=0, u(0)=uィイD20ィエD2,
Where F is a generalized play operator which auxiliary functions {fa, fd} are discontinuous. Our first aim is to construct solutions for (1), and to investigate their properties.
To avoid complication, this time we choose fa and fd as follows :
<<numerical formula>>
In this case, F is called delayed relay operator.
First we decompose (1) into
(2) uィイD2tィエD2-Δu+w=0
and
(3) wィイD2tィエD2+∂IィイD2uィエD2 (w) ∋ 0.
where, IィイD2uィエD2 (・) is a indicator function for the interval [fa(u), fa(u)], and ∂IィイD2uィエD2 is its subdifferential operator.
To prove the existence of solutions, we must construct a class of approximate solutions for (2) and (3), and show the compactness of those approximate solutions. As for (2), we can make use of a well-known energy estimate :
<<numerical formula>>
But as for (3), Sobolev technique is not useful because w is a step function. Therefore we make up the following plan :
(a) To derive the rate of variation of local length of contour curve u(・, t) = α of u per dt.
(b) to estimate |w|ィイD2BV(Ω)ィエD2 by |u|ィイD2c(Ω×(0,T))ィイD4-ィエD4ィエD2, |u|ィイD2WィイD11,2ィエD1(0,T;LィイD12ィエD1(Ω))ィエD2, and |u|ィイD2c([0,T];HィイD11ィエD1(Ω))ィエD2.
(c) To assure compactness of approximate solution of (3), by using BV norm.
(a)⇒(b) and (b)⇒(c) are obvious.
as for (a), we have the following partial result :
Let u be a solution of (2) with sufficiently smooth w, so that u itself is sufficiently smooth too. Let P and Q be two points on the contour curve u(・,t)=α. For small dt, the points P' and Q' defined by
<<numerical formula>>
are on the contour curve u(・,t+dt)=α, and satisfy the following estimate :
<<numerical formula>>
where PGィイD4→ィエD4=dx. This is the rate of variation of local length of contour curve u(・,t)=α per dt. It is still open how to estimate the right hand side of this inequality.
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