Lower Semicontinuity and Young Measures for Integral Functionals with Linear Growth

The first contribution of this thesis is a new proof of sequential weak* lower semicontinuity in $\mathrm{BV}(\Omega;\mathbb{R}^m)$ for integral functionals of the form \begin{align*} \mathcal{F}(u) &:= \int_\Omega f(x, \nabla u) \;\mathrm{d} x + \int_\Omega f^\infty \biggl(x, \frac{\mathrm{d} D^s u}{\mathrm{d} |D^s u|} \biggr) \;\mathrm{d} |D^s u| \\ &\qquad+ \int_{\partial\Omega} f^\infty \bigl( x, u|_{\partial \Omega} \otimes n_{\Omega} \bigr) \;\mathrm{d} \mathcal{H}^{d-1}, \qquad\qquad\qquad u \in \mathrm{BV}(\Omega;\mathbb{R}^m), \end{align*} where $f \colon \overline{\Omega} \times \mathbb{R}^{m \times d} \to \mathbb{R}$ is a quasiconvex Carath\'{e}odory integrand with linear growth at infinity, i.e.\ $|f(x,A)| \leq M(1+|A|)$ for some $M > 0$, and $f$ is such that the recession function $f^\infty(x,A) := \lim_{x'\to x,t\to\infty} t^{-1}f(x',tA)$ exists and is (jointly) continuous. In contrast to the classical proofs by Ambrosio \& Dal Maso [\textit{J.\ Funct.\ Anal.} 109 (1992), 76--97] and Fonseca \& M\"{u}ller [\textit{Arch.\ Ration.\ Mech.\ Anal.} 123 (1993), 1--49], the one presented here does not use Alberti's Rank One Theorem [\textit{Proc.\ Roy.\ Soc.\ Edinburgh Sect.\ A} 123 (1993), 239--274], but combines the usual blow-up method with a rigidity result for gradients. The second main result is the first general weak* lower semicontinuity theorem in the space $\mathrm{BD}(\Omega)$ of functions of bounded deformation for functionals of the form \begin{align*} \mathcal{F}(u) &:= \int_\Omega f(x, \mathcal{E} u) \;\mathrm{d} x + \int_\Omega f^\infty \biggl( x, \frac{\mathrm{d} E^s u}{\mathrm{d} |E^s u|} \biggr) \;\mathrm{d} |E^s u| \\ &\qquad + \int_{\partial \Omega} f^\infty \bigl( x, u|_{\partial \Omega} \odot n_\Omega \bigr) \;\mathrm{d} \mathcal{H}^{d-1}, \qquad\qquad\qquad u \in \mathrm{BD}(\Omega), \end{align*} where $f$ satisfies similar assumptions as before, in particular it is assumed symmetric-quasiconvex. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative $Eu$. The proof hinges on a construction of good blow-ups, which in turn is based on local rigidity arguments for some differential inclusions involving symmetrized gradients and an iteration of the blow-up construction. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in $\mathrm{BD}(\Omega)$, which is not available at present. This dissertation also presents an improved functional-analytic framework for generalized Young measures (with linear growth) as introduced by DiPerna \& Majda [\textit{Comm. Math. Phys.} 108 (1987), 667--689]. Localization principles and Jensen-type inequalities for such Young measures form the technical backbone of the lower semicontinuity proofs.