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The behavior of solutions to the biharmonic equation is well-understood insmooth domains. In the past two decades substantial progress has also been madefor the polyhedral domains and domains with Lipschitz boundaries. However, verylittle is known about higher order elliptic equations in the general setting.  In this paper we introduce new integral identities that allow to investigatethe solutions to the biharmonic equation in an arbitrary domain. We establish:(1) boundedness of the gradient of a solution in any three-dimensional domain;(2) pointwise estimates on the derivatives of the biharmonic Green function;(3) Wiener-type necessary and sufficient conditions for continuity of thegradient of a solution.

Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation