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In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator ${\Delta }^2$, on an arbitrary bounded Lipschitz domain $D$ in ${\mathbf{R}}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2\left(\partial D\right)$, and the normal derivative is in $L^2\left(\partial D\right)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2\left(\partial D\right)$.

We consider degenerated elliptic equations of the form $$\sum _{i,j}{D}_{x_i}(a_{ij}\left(x\right)D_{x_j}),\text{where}\lambda w(x\left)\right|\xi {|}^2\le \sum _{i,j}{a}_{ij}\left(x\right){\xi }_i{\xi }_j\le \Lambda w\left(x\right)|\xi {|}^2.$$ Under suitable assumptions on $w$, we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$. The main tool we use is an estimate for the Green function in terms of $w$.