Calculus of Variations and Geometric Measure Theory

L. Beck

Boundary regularity results for weak solutions of subquadratic elliptic systems

created by beck on 11 Nov 2009
modified on 17 Apr 2012

[BibTeX]

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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