Boundary value problems for degenerate elliptic-parabolic equations of the fourth order

Abstract:

We consider boundary value problems for the fourth-order linear equation \[ {A^{ijrs}}{u_{ijrs}} + {A^{ijr}}{u_{ijr}} + {A^{ij}}{u_{ij}} - \gamma {({a^{ij}}{u_i})_j} + {A^i}{u_i} + Fu = f\quad {\text {in}}\overline \Omega \] with smooth coefficients. The fourth-order part may degenerate on arbitrary subsets of $\overline \Omega$ i.e., ${A^{ijrs}}(x){m_{ij}}{m_{rs}} \geq 0$ for all $n \times n$ matrices $M$, with no restriction on where equality occurs. We assume the ${a^{ij}}$ part of the operator is uniformly elliptic (of second order) on $\Omega$ while $\gamma$ is a parameter allowing us to increase modulus of ellipticity as much as needed. As in Fichera’s second-order elliptic-parabolic equations [see, for example, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. (8) 5 (1956), 1-30], because of the degeneracy, there may be characteristic portions of the boundary; however, we restrict our attention to the noncharacteristic case. We define a weak solution to the Dirichlet problem and obtain existence and uniqueness results. The question of regularity is addressed; elliptic regularization is used to obtain a Sobolev-type global regularity result. The equation models an anisotropic, inhomogeneous plate under tension that 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.