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- W2069816462 abstract "Let Q be a bounded smooth domain in Rn, and suppose the singlelayer potential of aQ coincides for y 0 Q with the function clyI-I (c > 0) . Then aQ is a sphere centered at the origin. Throughout this paper we assume Q c 23 (the case Rn is similar) is a bounded domain, and the boundary aQ is smooth enough that for (1) JAQ~~u(y IX Y VY E R one has a u au (2) 4-r Ot onaQ, where dau denotes the surface measure, + indicates the limit from the exterior and the limit from the interior (see [K, p. 164]), and n is the outward normal vector to a Q, which we assume to exist. If Q has this property, then we say Q is smooth. Theorem. Let Q be a bounded smooth domain in R3, and suppose for some c > O (3) jda -,, Vy $4Q. (3) /9~~~~~Q 1X Y IYI Then, aQ is a sphere centered at the origin. Proof. Since the single-layer potential is continuous in 23 (see [K, p. 160]) (3) implies 0 E Q. Therefore we can take two balls B1, B2 both centered at the origin, such that B1 is the largest ball in Q and B2 is the smallest ball containing Q. Now the idea is to show that aBI = aB2 = aQ. Now let y E aB1 n Q and y2 E aB2 n OQ, then 1y21 > ylvI. Thus it suffices to prove Iy221 < Iyl v . Define u to be the function represented by ( 1). Then by the maximum principle (since u is harmonic in Q) max u in Q is attained on Received by the editors January 7, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 3 1B20." @default.
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- W2069816462 date "1992-04-01" @default.
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- W2069816462 title "A characterization of the sphere in terms of single-layer potentials" @default.
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- W2069816462 doi "https://doi.org/10.1090/s0002-9939-1992-1162956-7" @default.
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