Directionally Dependent Asymptotic Behavior of Biharmonic Functions with Applications to Elasticity
Kenneth B. Howell · SIAM Journal on Mathematical Analysis · 1985
The behavior of biharmonic functions defined on infinite domains is investigated with interest focused on obtaining local bounds on the gradients of. these functions based on assumed local bounds on the original biharmonic functions. The assumed bounds involve two independent distances, each raised to some arbitrarily chosen exponent. One of the two distances is the distance to some fairly arbitrary subset of the closure of the domain (e.g., the boundary) while the other is the distance to some arbitrarily chosen plane. The derived bounds on the gradients reflect this “directional dependency”. In addition, as the first distanceincreases, the derived bounds on the gradients tend to either increase more slowly or decrease more rapidly than the assumed bounds on the original biharmonic functions. Several classes of problems from classical elasticity are then discussed. These problems involve unbounded domains and either periodic or “slightly periodic” boundary data. Using the results from the first part of the paper “physically reasonable” assumptions are shown to insure appropriate periodicity in the solutions to periodic boundary value problems and appropriate uniqueness in the solutions to “slightly periodic” boundary value problems.