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Description:
This thesis is concerned with an important class of quasilinear elliptic equations: the p-Poisson equations -div(|\nabla u|^{p-2} \nabla u) = f in Ω, where 1 [ p [ infty and Ω denotes a bounded Lipschitz domain in R^d, d]=2. Equations of this type appear, inter alia, in various problems in continuum mechanics, for instance in the mathematical modelling of non-Newtonian fluids. Furthermore, the p-Poisson equations possess a certain model character for more general quasilinear elliptic problems. The central aspect of this thesis is the regularity analysis of solutions u to the p-Poisson equation in the so-called adaptivity scale B^σ_τ(L_τ(Ω)), 1/τ = σ/d + 1/p, σ ] 0, of Besov spaces. It is well-known that the smoothness parameter σ determines the approximation rate of the best n-term wavelet approximation, and hence provides information on the maximal convergence rate of certain adaptive numerical wavelet methods. To derive Besov regularity estimates for solutions to the p-Poisson equation, two approaches are pursued in this work. The first approach makes use of the fact that under appropriate conditions the solutions to the p-Poisson equation admit certain higher regularity in the interior of the domain, in the sense that they are locally Hölder continuous. In general, the Hölder semi-norms may explode as one approaches the boundary of the domain, but this singular behavior can be controlled by some power of the distance to the boundary. It turns out that the combination of global Sobolev regularity and locally weighted Hölder regularity can be used to derive Besov smoothness in the adaptivity scale for solutions to the p-Poisson equation. The results of the first approach are stated in two steps. At first, a general embedding theorem is proved, which says that the intersection of a classical Sobolev space with a Hölder space having the above mentioned properties can be embedded into certain Besov spaces in the adaptivity scale. The proof of this result is based on extension arguments in connection with the ...