An existence and uniqueness theorem for the two-dimensional linear membrane shell equations. (English) Zbl 0856.73038

Two-dimensional membrane equations of a linear elastic clamped shell are considered. The middle surface is uniformly elliptic, i.e. its two principal radii of curvature are uniformly positive and bounded. This assumption allows to eliminate the normal component of the displacement of the middle surface.
The reduced equations are tackled by the Lax-Milgram lemma. To this end, the bilinear form defined by the elliptic operator is verified to be an inner product in \(H^1_0\), the space of solutions. A lower bound is obtained for the corresponding quadratic form, which implies that the form vanishes at solutions of a linear elliptic system with zero data. It is shown that the latter system has only zero solution provided the middle surface is analytic. As a result, the unique solvability is proved.


74K15 Membranes
74K20 Plates
35Q72 Other PDE from mechanics (MSC2000)