Singularities

Abstract

All of the existence theorems for minimizers of integral functionals defined on Sobolev spaces $\mathrm {W}^{1,p}(\varOmega ;\mathbb {R}^m)$ that we have seen so far required that $p > 1$. Extending the existence theory to the linear-growth case $p=1$ turns out to be quite intricate and necessitates the development of new tools.

Appendices

Notes and Historical Remarks

The Reshetnyak Continuity Theorem 10.3 was first proved in [226], whereas our argument is from [15]. The paper [94] contains more on different notions of convergence for measures.

Often, tangent measures are only considered as defined on the unit ball instead of the whole of the space (see, for example, Section 2.7 in [15]). This is, however, sometimes restrictive. Here, we use Preiss’s original theory as developed in [224], also see Chapter 14 of [183]. The original definition, however, explicitly excluded the zero-measure from ${{\mathrm{Tan}}}(\mu , x_0)$, which here we include. Proposition 10.5 is a slight improvement of Preiss’ existence theorem for non-zero tangent measures, see Theorem 2.5 in [224] or the appendix of [227]. The proof of Lemma 10.4 is adapted from Theorem 2.44 in [15]. Lemma 10.6 is originally due to Larsen, see Lemma 5.1 in [174].

We remark that the study of local properties of a measure via its tangent measures has its limits. Most strikingly, Preiss constructed a purely singular positive measure on a bounded interval (in particular a $\mathrm {BV}$-derivative) such that each of its tangent measures is a multiple of Lebesgue measure, see Example 5.9 (1) in [224]. Also see [219] for a measure that has every local measure as a tangent measure at almost every point.

The notion of area-strict convergence in $\mathrm {BV}$ seems to be somewhat less well known than it deserves. However, as shown in the next chapter, it is the right one when considering integral functionals.

Alberti’s original proof [4] of what is now called Alberti’s rank-one theorem is via a “decomposition technique” together with the $\mathrm {BV}$-coarea formula; a streamlined version of his proof can be found in [90]. Another proof in two dimensions was announced in [5]. There is also now a nice short geometric proof [181]. Our argument is more in the spirit of PDE theory and has several other implications, see [92]. For more on the wave cone we refer to [98, 209, 210, 229, 267, 268].

The Kirchheim–Kristensen Theorem 10.13 was already announced in 2011 [161] with a simpler proof in a special case. The full proof appeared in [162]. The theorem actually holds in more generality, see Problem 10.10.

Problems

10.1.

Let $\varLambda \in \mathscr {M}^+(B(0,1))$ be a finite and positive Borel measure. Then, we have $\varLambda (\partial B(0,r)) = 0$ for all but finitely many $r \in (0,1)$. Hint: Consider the sets $E_n := \{\, r \in (0,1) \ \,\mathbf : \ \, \varLambda (\partial B(0,r)) > 1/n \,\}$, where $n \in \mathbb {N}$.

10.2.

Prove (10.8) using the integration-by-parts definition of $\mathrm {BV}$-functions.

10.3.

Let $\varOmega \subset \mathbb {R}^d$ be open.

• (i) Show Reshetnyak’s lower semicontinuity theorem: Let $f :\varOmega \times \mathbb {S}^{N-1} \rightarrow [0,\infty ]$ be lower semicontinuous and convex in the second argument. For any sequence $(\mu _j) \subset \mathscr {M}(\varOmega ;\mathbb {R}^N)$ with $\mu _j \overset{*}{\rightharpoonup }\mu$ it holds that

  $$ \int f \biggl ( x, \frac{\mathrm {d}\mu }{\mathrm {d}|\mu |}(x) \biggr ) \;\mathrm {d}|\mu |(x) \le \liminf _{j\rightarrow \infty } \int f \biggl ( x, \frac{\mathrm {d}\mu _j}{\mathrm {d}|\mu _j|}(x) \biggr ) \;\mathrm {d}|\mu _j|(x). $$

  Hint: Adapt the strategy of the proof of Theorem 10.3.

• (ii) Let $(\mu _j) \subset \mathscr {M}(\varOmega ;\mathbb {R}^N)$ with $\mu _j \rightarrow \mu$ area-strictly, i.e., $\mu _j \overset{*}{\rightharpoonup }\mu$ and $\langle \mu _j \rangle (\varOmega ) \rightarrow \langle \mu \rangle (\varOmega )$. Show that then also $\mu _j \rightarrow \mu$ strictly. Hint: If $\mu = a \mathscr {L}^d + \mu ^s$ is the Lebesgue–Radon–Nikodým decomposition of $\mu$, then define $\tilde{\mu } \in \mathscr {M}(\varOmega ;\mathbb {R}^N \times \mathbb {R})$ as $\tilde{\mu }(B) := (a, 0) \mathscr {L}^d + (0,1) \mu ^s$ and use the Reshetnyak continuity theorem.

• (iii) Show that the mapping $\mu \in \mathscr {M}(\varOmega ;\mathbb {R}^N) \mapsto \langle \mu \rangle (\varOmega )$ is weakly* lower semicontinuous.

10.4.

Let $u \in \mathrm {BV}(\varOmega ;\mathbb {R}^m)$ and assume that $x_0 \in \varOmega$ is such that

$$ \frac{\mathrm {d}D^s u}{\mathrm {d}|D^s u|}(x_0) = a \otimes n \qquad \text {for some}\, a \in \mathbb {S}^{m-1} , n \in \mathbb {S}^{d-1}. $$

Set

$$ u_r(y) := \frac{r^d}{|Du|(Q_n(x_0,r))} \cdot \frac{u(x_0 + ry) - [u]_{B(x_0,r)}}{r}, \qquad y \in Q_n(x_0,r), \; r > 0, $$

where $Q_n(x_0,r)$ is an open cube with midpoint $x_0 \in \varOmega$, side-length r, and two faces orthogonal to n. Then, show that for $|D^s u|$-almost every such $x_0$ it holds that $u_r \overset{*}{\rightharpoonup }u_0$ in $\mathrm {BV}(Q_n(0,1);\mathbb {R}^m)$ with $u_0(y) = a \psi (y \cdot n)$ for a bounded function $\psi :(-1/2,1/2) \rightarrow \mathbb {R}$. Further show that $\psi$ can be chosen to be increasing. Hint: Use Corollary 10.8.

10.5.

Let $u \in \mathrm {BV}(\varOmega ;\mathbb {R}^m)$. Show that for $|Du|$-almost every $x_0 \in \varOmega$, every $\tau \in {{\mathrm{Tan}}}(Du, x_0)$ is a $\mathrm {BV}$-derivative, $\tau = Dv$ for some $v \in \mathrm {BV}_\mathrm {loc}(\mathbb {R}^d;\mathbb {R}^m)$, and with $P_0 := \frac{\mathrm {d}Du}{\mathrm {d}|Du|}(x_0)$ it holds that:

• (i) If ${{\mathrm{rank}}}~P_0 \ge 2$, then $v(x) = v_0 + \alpha ~P_0 x$, where $\alpha \in \mathbb {R}$, $v_0 \in \mathbb {R}^m$.

• (ii) If $P_0 = a \otimes n$ ($a \in \mathbb {R}^m$, $n \in \mathbb {S}^{d-1}$), then there exist $h \in \mathrm {BV}(\mathbb {R})$, $v_0 \in \mathbb {R}^m$ such that $v(x) = v_0 + h(x \cdot n)a$.

What does Alberti’s Rank-One Theorem imply about (i)? This problem is continued in Problem 12.7.

10.6.

Let $\varOmega \subset \mathbb {R}^d$ be an open set and let $u \in \mathrm {L}^1(\varOmega ;\mathbb {R}^m)$ with the property that the r’th (weak) derivative $D^r u \in \mathscr {M}(\varOmega ;\mathrm {SLin}^r(\mathbb {R}^d;\mathbb {R}^m))$ exists for some $r \in \mathbb {N}$, where $\mathrm {SLin}^r(\mathbb {R}^d;\mathbb {R}^m)$ contains all symmetric r-linear maps from $\mathbb {R}^d$ to $\mathbb {R}^m$. Then show that for $|(D^r u)^s|$-almost every $x\in \varOmega$ there exist $a(x)\in \mathbb {R}^m \setminus \{0\}$, $n(x)\in \mathbb {S}^{d-1}$ such that

$$ \frac{\mathrm {d}(D^r u)^s}{\mathrm {d}|(D^r u)^s|}(x) = a(x) \otimes \underbrace{n(x) \otimes \cdots \otimes n(x)}_{r\,\,\text {times}}. $$

Here, the tensor on the right-hand side is the r-linear map

$$ V(v_0,v_1,\ldots , v_r) := (a(x) \cdot v_0) \prod _{i=1}^r (n(x) \cdot v_i), \qquad v_0 \in \mathbb {R}^m, \; v_1,\ldots , v_r \in \mathbb {R}^d. $$

This is Alberti’s theorem for higher-order gradients [4]. Hint: Use Theorem 10.10.

10.7.

Let $\mu \in \mathscr {M}(\varOmega ;\mathbb {R}^{d\times d})$ be a matrix-valued measure such that (in the distributional, i.e. $\mathrm {C}^\infty _c(\varOmega )^*$-weak sense)

$$ {{\mathrm{div\,}}}\mu \in \mathscr {M}(\varOmega ; \mathbb {R}^d). $$

Prove that then

$$ {{\mathrm{rank}}}\biggl ( \frac{\mathrm {d}\mu }{\mathrm {d}|\mu |}(x) \biggr ) \le d-1 \qquad \text {for} \,|\mu ^s| {\text {-a.e.}} \,\, x\in \varOmega . $$

10.8.

A map $u \in \mathrm {L}^1(\varOmega ;\mathbb {R}^d)$ ($\varOmega \subset \mathbb {R}^d$ a bounded Lipschitz domain) is called a function of bounded deformation  if the symmetric part of its distributional derivative is a measure,

$$ Eu := \frac{Du+(Du)^T}{2}\in \mathscr {M}(\varOmega ;\mathbb {R}^{d \times d}_\mathrm {sym}). $$

We collect all these functions into the set $\mathrm {BD}(\varOmega )$; see [12, 273, 274] for the theory of this space. Let $\mu = (\mu ^k_j) \in \mathscr {M}(\varOmega ,\mathbb {R}^{d \times d}_\mathrm {sym})$ be the symmetrized derivative of some $u \in \mathrm {BD}(\varOmega )$, $\mu = Eu$. Verify that then

$$ 0 = \mathscr {A}\mu := \biggl [ \sum _{i=1}^d \partial _i \partial _k \mu _{i}^j+\partial _i \partial _j \mu _{i}^k-\partial _j \partial _k \mu _{i}^i-\partial _i \partial _i \mu _{j}^k \biggr ]_{j, k=1,\ldots , d} $$

in the sense of $\mathrm {C}^\infty _c(\varOmega )^*$. These equations are often called the Saint-Venant compatibility conditions in applications.

10.9.

In the situation of the previous problem and denoting the Lebesgue–Radon–Nikodým decomposition of the symmetrized derivative Eu of $u \in \mathrm {BD}(\varOmega )$ by

prove that for $|E^s u|$-almost every $x \in \varOmega$, there exist $a(x), b(x)\in \mathbb {R}^d \setminus \{0\}$ such that

$$ \frac{\mathrm {d}E^s u}{\mathrm {d}|E^s u|}(x) = a(x) \odot b(x), $$

where we define the symmetrized tensor product as $a \odot b := (a\otimes b+b\otimes a)/2$ for $a, b\in \mathbb {R}^d$. Hint: Use Theorem 10.10 and the previous problem. This result was one of the main motivations for Theorem 10.10.

10.10.

Assume that $D \subset \mathbb {R}^N$ is a balanced cone, that is, for $v \in D$ and $t \in \mathbb {R}$ also $tv \in D$, and further assume that ${{\mathrm{span}}}D = \mathbb {R}^N$. Let $h^\infty :\mathbb {R}^N \rightarrow \mathbb {R}$ be positively 1-homogeneous and convex in all directions in D. Show that $h^\infty$ is convex at every $F \in D$, i.e., there exists an affine function $a_F :\mathbb {R}^N \rightarrow \mathbb {R}$ with

$$ h^\infty (F) = a_F(F) \qquad \text {and}\qquad h^\infty \ge a_F. $$

This is (almost) the main result of [162].