Ph.D. Thesis
Inserted: 11 nov 2009
Last Updated: 17 apr 2012
Year: 2008
Abstract:
The current thesis makes a contribution to the field of regularity theory of second-order nonlinear elliptic systems. We consider weak solutions $u \in g + W^{1,p}_0(\Omega,R^N)$ of the inhomogeneous elliptic system \[ - \rm{div } a( \, \cdot \,, u, Du) \, = \, b( \, \cdot \,,u,Du) \qquad \text{in } \Omega \] with prescribed boundary data $g \in W^{1,p}(\Omega,R^N)$, a bounded domain $\Omega \subset R^n$ of class $C^1$ and a vector field $a(\cdot,\cdot,\cdot)$ which satisfies standard continuity, ellipticity and growth conditions. The inhomogeneity $b: \overline{\Omega} \times R^N \times R^{nN} \to R^N$ is assumed to be a Carathéodory function obeying either a controllable or a natural growth condition. Under these assumptions, the following higher integrability and regularity results (up to the boundary of $\Omega$) are achieved, mainly for the subquadratic case $1<p<2$:
We first require that $\Omega$ and $g$ are of class $C^{1,\alpha}$, $\alpha \in (0,1)$, and that the coefficients are Hölder continuous with exponent $\alpha$ with respect to the first and second variable. Via the method of $\mathcal{A}$-harmonic approximation we give a characterization of regular points for $Du$ up to the boundary which extends known results to the inhomogeneous case. The proof yields directly the optimal higher regularity on the regular set (i.\,e., local Hölder continuity of $Du$ with exponent $\alpha$).
Provided that the boundary data $g$ is of class $C^1$ and that the coefficients are uniformly continuous we then show Calderón-Zygmund estimates, a higher integrability result that yields, in contrast to classical higher integrability obtained from the application of Gehring's Lemma, a quantified gain in the higher integrability exponent. If the coefficients do not depend explicitly on $u$ and if the inhomogeneity $b(x,u,z) \equiv b(x)$ belongs to $L^{p/(p-1)}$, then there holds: $b \in L^{q/(p-1)}(\Omega,R^N)$ and $g \in W^{1,q}(\Omega,R^N)$ imply $Du \in L^q(\Omega,R^{nN})$ for $q \in [p,\frac{np}{n-2} + \delta_1]$ (or $q$ arbitrary if $n=2$).
Moreover, in low dimensions $n \in (p,p+2]$, we prove via the direct method and Morrey-type estimates: $u$ is locally Hölder continuous with every exponent $\lambda \in (0,1-\frac{n-2}{p})$ outside a singular set of Hausdorff dimension less than $n-p$. This result holds true both for non-degenerate and degenerate systems.
The last part of the thesis is devoted to techniques which allow us to estimate the Hausdorff dimension of the singular set of $Du$ in $\overline{\Omega}$. Here, all the result achieved so far are of importance. Assuming that $\Omega$ and $g$ are of class $C^{1,\alpha}$ for some $\alpha \in (0,1)$ and that the coefficients are Hölder continuous with exponent $\alpha$ with respect to the first and second variable, we find: The Hausdorff dimension of the singular set of $Du$ does not exceed $\min\{n-p,n-2\alpha\}$ whenever $n \in (p,p+2]$. In particular, for $\alpha > \frac{1}{2}$ this implies that almost every boundary point is in fact a regular one (for a natural growth condition this is proved only for $p=2$). Furthermore, this conclusion remains valid for coefficients of the form $a(x,u,z) \equiv a(x,z)$ and inhomogeneities of controllable growth without any restriction on the dimension $n$. The proof is based on finite difference operators, interpolation techniques and fractional Sobolev spaces. To extend this strategy up to the boundary, we present two different methods: for controllable growth we proceed directly and use a family of comparison maps (which are solutions of some regularized system) as well as Calderón-Zygmund estimates. For natural growth, however, we argue in a direct way and employ the fact that slicewise mean values of the coefficients are weakly differentiable in the normal direction.
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