On the existence of boundary limit values for solutions to polyharmonic equations

Two criteria for the existence of limit values of a polyharmonic function on the boundary of ball are obtained by considering the function belonging to the class and having no limit values on the boundary of the domain. The necessary and sufficient conditions for the function are specified to have values on the boundary in terms of quadratic mean and extended the results to the multidimensional case. A family of traces on the spheres are considered concentric to the boundary for any continuous function on the ball. The results show that a harmonic function has a limit on the boundary if and only if it is compact and the harmonic function has a weak limit on the boundary if and only if it is weakly compact. It is also found that a harmonic function has a limit on the boundary if and only if it is bounded. These results found the conditions and the values of boundedness for different harmonic functions as a limit.