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THE TRACE THEOREM ON ANISOTROPIC SOBOLEV ... - Project Euclid
In this note, we give the trace theorem on anisotropic Sobolev spaces which ... Hl(Rn+) coincides with HlÎn(Rn+) defined by Bardos and Rauch [1]. For m>2, we ...
Tόhoku Math. J. 46 (1994), 393^01
THE TRACE THEOREM ON ANISOTROPIC SOBOLEV SPACES Dedicated to Professor Takesi Kotake on his sixtieth birthday
MAYUMI OHNO, YASUSHI SHIZUTA AND TAKU
YANAGISAWA
(Received April 28, 1993, revised June 10, 1993) Abstract. The trace theorem on anisotropic Sobolev spaces is proved. These function spaces which can be regarded as weighted Sobolev spaces are particularly important when we discuss the regularity of solutions of the characteristic initial boundary value problem for linear symmetric hyperbolic systems.
In this note, we give the trace theorem on anisotropic Sobolev spaces which appear in the study of the initial boundary value problem for linear symmetric hyperbolic systems with characteristic boundary. The function spaces with which we concern ourselves will be denoted by H™(Ω), Ω being an open set in Rn lying on one side of its boundary. Denoting the usual Sobolev space by Hm(Ω), we have the continuous embedding Hm(Ω) ci+ Hζ(Ω) for ra = 0, 1 , . . . . H™(Ω) is anisotropic in the sense that the tangential derivatives and the normal derivatives are treated in different ways in this space. In contrast with the case where the boundary is non-characteristic, the solution of the characteristic initial boundary value problem for symmetric hyperbolic systems lies in general in Hζ(Ω), not in Hm(Ω). The trace theorem on H™(Ω) is needed especially when we consider the compatibility condition. This is the motivation for the present work. For simplicity, we suppose that Ω is a half-space in Rn. Let Rn+ =
{{t9y)\t>09yeR"-1}.
Let p G C°°(Λ+) be a monotone increasing function such that ρ(ί) = t in a neighborhood of the origin and ρ(t) = 1 for any t large enough. By means of this function, we define the differential operator in the tangential directions
