| We consider boundary value problems for the self-adjoint operator Lu:= (A{dollar}sp{lcub}rm ijrs{rcub}{dollar}u,{dollar}sb{lcub}rm ij{rcub}{dollar}),{dollar}sb{lcub}rm rs{rcub} - gamma{dollar}(a{dollar}sp{lcub}rm ij{rcub}{dollar}u,{dollar}sb{lcub}rm i{rcub})sb{lcub}rm j{rcub}{dollar} with smooth coefficients. The fourth order part may degenerate on arbitrary subsets of {dollar}bar Omega{dollar}. i.e., A{dollar}sp{lcub}rm ijrs{rcub}{dollar}(x){dollar}etasb{lcub}rm ij{rcub}etasb{lcub}rm rs{rcub}{dollar} {dollar}geq{dollar} 0 for all symmetric matrices {dollar}eta{dollar}, with no restriction on where equality occurs. We assume the second order part is uniformly elliptic on {dollar}Omega{dollar}. As in Fichera's second order elliptic-parabolic equations (see, e.g., Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Acc. Naz. Lincei Mem., Ser. 8, v 5, 1956, pp. 1-30), due to the degeneracy, boundary conditions are not prescribed on all of {dollar}partialOmega{dollar}. For the well-posed "Dirichlet problem" one specifies u on {dollar}partialOmega{dollar} and {dollar}{lcub}partial{lcub}rm u{rcub}{rcub}over{lcub}partial{lcub}rm n{rcub}{rcub}{dollar} only on {dollar}Sigmasp{lcub}*{rcub}{dollar}:={dollar}partialOmega{lcub}{dollar} sc X {dollar}inpartialOmega{dollar}: A{dollar}sp{lcub}rm ijrs{rcub}{dollar}(x)n{dollar}sb{lcub}rm i{rcub}{dollar}n{dollar}sb{lcub}rm j{rcub}{dollar}n{dollar}sb{lcub}rm r{rcub}{dollar}n{dollar}sb{lcub}rm s{rcub}{dollar} = 0{dollar}{rcub}{dollar}. We define a weak solution to the problem by: B(u,v) = (f,v) {dollar}forall{dollar} sc V {dollar}in{lcub}bf {lcub}cal H{rcub}{rcub}{dollar} where B({dollar}cdot{dollar},{dollar}cdot{dollar}) is the bilinear form associated with the operator and {dollar}{lcub}bf {lcub}cal H{rcub}{rcub}{dollar} is an appropriate Hilbert space. The existence and uniqueness of the weak solution are demonstrated in the usual way, and the question of regularity is addressed. Following the work of Oleinik (Oleinik & Radkevic, Second Order Equations with Nonnegative Characteristic Form. AMS & Plenum, 1971), elliptic regularization is used to obtain a Sobolev-type global regularity result in cases when the operator degenerates at the boundary as well as with no boundary degeneracy. The results in the case of degeneracy at the boundary are obtained using estimates similar to those of Kohn & Nirenberg (Degenerate Elliptic-Parabolic Equations of the Second Order, Comm. Pure & Appl. Math., v20, pp. 797-872). The equation models an anisotropic, inhomogeneous, hyperelastic plate under tension which can lose stiffness at any point and in any direction. The regularity result has the satisfying physical interpretation that sufficient tension results in a smooth solution. |
