Bounds in the Cauchy Problem for a Fourth Order Quasi-Linear Equation

Abstract

Consider the Cauchy problem for the fourth order quasi-linear equation \[ \Delta ^2 v = h( {x,v,v_{,i} ,\Delta v,\nabla v_{,i} } ) \]where $\Delta $ is the Laplace operator, the comma notation denotes partial differentiation with respect to $x_i $, h satisfies a uniform Lipschitz condition in all but the x-variables, and upper bounds for the error in measurement of the Cauchy data are known. Pointwise bounds for the solution and the square of the gradient are obtained by means of an a priori inequality which is derived under the additional assumption that the solution is uniformly bounded in the domain under consideration. These bounds can be improved by the Ritz method. The technique can be utilized to obtain pointwise estimates in the Cauchy problem for certain coupled systems of elliptic partial differential equations.