In this paper a method of Bramble and Payne [4], [5], and [7] is used to obtain a. priori point wise bounds at interior points for the solution of (i) the Dirichlet problem for a rather general second order nonlinear parabolic operator;(ii) the first boundary value problem for certain linear fourth order elliptic operators and (iii) the first initial-boundary value problem for certain linear fourth order parabolic operators. In addition, a priori bounds for the energy integrals corresponding to the fourth order elliptic operators are derived. Since the bounds obtained are in terms of La integrals of the data (and, for the fourth order operators, derivatives of the data) the Rayleigh-Ritz technique may be employed in the linear problems to obtain close bounds.
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