Multiple Solutions of H-Systems and Rellich’s Conjecture

where H > 0 is a given constant, y : dR + R3 is a given function and r c R3 is a given oriented Jordan curve. If u is a solution of (H)-(P), then u ( a ) represents a “soap bubble”, that is, a surface with mean curvature H (at all points x c R where V u ( x ) # O ) spanning r. Let us assume that y(aR) (respectively r) is contained in a closed ball of radius R. It was proved by S. Hildebrandt [8] that both the Dirichlet and Plateau problems have at least one solution if HR 4 1 (this was an improvement over earlier results of Heinz [6] and Werner [19]). Moreover this result is sharp when I‘ is a circle: there is no solution of (H)-(P) if HR > 1 (see Heinz [7]). In case r is a circle of radius R and H R c 1 , it is easy to check that there exist two solutions of (H)-(P), namely: 1. the “small” spherical bubble B, of curvature H spanned by r, 2. the “large” spherical bubble Bz of curvature H spanned by r; see Figure 1).