Let $u$ be harmonic in a bounded domain $D$ with smooth boundary. We prove that if the boundary values of $u$ belong to ${L}^{p}\left(\sigma \right)$, where $p\ge 2$ and $\sigma $ denotes the surface measure of $\partial D$, then it is possible to approximate $u$ uniformly by function of bounded variation. An example is given that shows that this result does not extend to $p\<2$.

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 ${L}^{2}\left(\partial D\right)$.

The purpose of this paper is to study nonnegative solutions u of the nonlinear evolution equations
∂u/∂t = Δφ(u), x ∈ R^{n}, 0 < t < T ≤ +∞ (1.1)
Here the nonlinearity φ is assumed to be continuous, increasing with φ(0) = 0. This equation arises in various physical problems, and specializing φ leads to models for nonlinear filtrations, or for the gas flow in a porous medium. For a recent survey in these equations...

The purpose of this work is to study the class of non-negative continuous weak solutions of the non-linear evolution equation
∂u/∂t = ∆φ(u), x ∈ R^{n}, 0 < t < T ≤ +∞.

Let $L$ be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in ${L}^{p}$ norm between the maximal function and the square function of solutions to $L$ in Lipschitz domains. Several applications of this result are discussed.

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