An existence theorem for the Dirichlet problem in the elastodynamics of incompressible materials
The authors consider initial-boundary value problems. The incompressibility condition causes difficulties in dealing with boundary conditions and several authors attempted various approximations of such problems by solutions for compressible materials. The authors by-pass energy estimates in terms of spatial derivatives by replacing them by derivatives with respect to time. Using the notation \(F^ i_{\alpha}=\partial y^ i/\partial x^{\alpha}\), where y denotes the position of a point inside the material, the equations of motion are \[ \ddot y^ i=(-\partial p/\partial x^{\alpha})(\partial x^{\alpha}/\partial y^ i)+(\partial^ 2W(F)/\partial F^ i_{\alpha}\partial F^ i_{\beta})(\partial^ 2y^ j/\partial x^{\alpha}\partial x^{\beta})+f^ i, \] where \(W=W(F)\) is the stored energy function, \(p=p(x,t)\) is an unknown pressure, and \(f=f(x,t)\) is a prescribed body force. The motion must satisfy the constraint det F\(=1\). The initial- boundary conditions are: \(y(x,0)=y_ 0(x)\), \(\dot y(x,0)=y_ 1(x)\) in \(\Omega\), and \(y(x,t)=x\) for all \(t>0\), \(x\in \partial \Omega\). To force uniqueness the authors normalize the pressure distribution: \(\int_{\Omega}p(x,t)dx=0.\) The authors make several assumptions concerning smoothness and compatibility of data (such as vanishing of second and third order time derivatives on the boundary), which permit them to prove the existence and regularity of q and p (of order 4-k on [0,T] and k in \(\Omega\), where k is the order of smoothness of f(x,t): \(f\in \cap^{2}_{k=0}C^ k([0,T];H^{2-k}(\Omega))\). The technique of proof is outlined. It requires solvability of an auxiliary linear elliptic system. Under additional assumptions they prove uniqueness of solutions.
local (in time) existence of classical solutions to Dirichlet problem
limits of solutions for compressible materials
analogues of energy for sufficiently high derivatives of solution
scientific article; zbMATH DE number 4078905
scientific article