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We prove $hp$-optimal error estimates for interior penalty discontinuousGalerkin methods (IPDG) for the biharmonic problem with homogeneous essentialboundary conditions. We consider tensor product-type meshes in two and threedimensions, and triangular meshes in two dimensions. An essential ingredient inthe analysis is the construction of a global $H^2$ piecewise polynomialapproximants with $hp$-optimal approximation properties over the given meshes.The $hp$-optimality is also discussed for $\mathcal C^0$-IPDG in two and threedimensions, and the stream formulation of the Stokes problem in two dimensions.Numerical experiments validate the theoretical predictions and reveal that$p$-suboptimality occurs in presence of singular essential boundary conditions.

$hp$-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem