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Volume :24 Issue : 2 1997      Add To Cart                                                                    Download

High order regularity of the singularities of the solution of a parabolic equation in a singular domain

Auther : BOUBAKER-KHALED SADALLAH

 Department of Mathematics, Ecole Normale Supêrieure, 16050-Kouba, Algiers, Algeria
 
(Note: Mathematical characters in this abstract are not shown properly in html format. Please send us an e-mail if you are interested in this paper)

 ABSTRACT

We are concerned, in this work, with the parabolic equation:

(*)

in the following non convex polygonal domain ) described by the variables (t, x) e D

= (—1, 0) x (—1, 1) U [ 1) x (0, 1).

The boundary conditions are of Cauchy—Dirichlet type while the second member f of the equation lies in the non symmetric Sobolev space H” “(1 defined, for p e %J, by:

denotes the well known Lebesgue space.

.. satisfies some compatibility conditions, the “natural” space of solutions, in case of . enough (e.g., convex domain) is the Sobolev space “ 4 1)( According to some

‘ we know that the space of solutions may be different. In Sadallah (1996), we dealt with

 (*) for p = 0. Now, we are interested in the optimal regularity of the singularities .. appear in the solution when p is any integer.

. main result of this work shows the existence of 2(p + 1) singularities (vi, k)j=O, k=O p :: for all f e H” 4 satisfying some compatibility conditions, the solution belongs

1 4 l)(ç IRv k

in addition , the singularities fulfill the optimal regularity condition, for all j = 0, 1 and

V . k Hnj+k 4(rJ+k)( r . < 2j

 

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