Let be harmonic in a bounded domain with smooth boundary. We prove that if the boundary values of belong to , where and denotes the surface measure of , then it is possible to approximate uniformly by function of bounded variation. An example is given that shows that this result does not extend to .
In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator , on an arbitrary bounded Lipschitz domain in . We establish existence and uniqueness results when the boundary values have first derivatives in , and the normal derivative is in . The resulting solution takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of is shown to be in .
The purpose of this paper is to study nonnegative solutions u of the nonlinear evolution equations ∂u/∂t = Δφ(u), x ∈ Rn, 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 ∈ Rn, 0 < t < T ≤ +∞.
Let be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in norm between the maximal function and the square function of solutions to in Lipschitz domains. Several applications of this result are discussed.
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