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Approximation of harmonic functions

Björn E. J. Dahlberg — 1980

Annales de l'institut Fourier

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 ( σ ) , where p 2 and σ denotes the surface measure of 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 .

The Dirichlet problem for the biharmonic equation in a Lipschitz domain

Björn E. J. DahlbergC. E. KenigG. C. Verchota — 1986

Annales de l'institut Fourier

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

Non-negative solutions of generalized porous medium equations.

Bjorn E. J. DahlbergCarlos E. Kenig — 1986

Revista Matemática Iberoamericana

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...

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