Bounded discontinuous elliptic coefficients need not give $H^2$ solutions

Given: The interval $\mathrm{\Omega }=(-1,1)$; the coefficient $a(x)=\{\begin{array}{ll}1,& x<0,\\ 2,& x>0,\end{array}$ the primitive-shaped function $u(x)=\{\begin{array}{ll}x,& x<0,\\ x\mathrm{/}2,& x>0,\end{array}$ and the operator $Lu=-(a{u}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, so that $a^{11}=a$ and $b=c=0$.

A class $u\in H^1(-1,1)$ is a local weak solution of $Lu=0$ on $(-1,1)$ if ${\int }_{-1}^{1}a{u}^{\mathrm{\prime }}\overline{v^{\mathrm{\prime }}}dx=0$ for every $v\in C_c^{\mathrm{\infty }}(-1,1)$. (Local weak solutions of a divergence-form operator, Uniformly elliptic divergence-form operators and their sesquilinear forms)

The coefficient $a$ is measurable and bounded with $1\le a\le 2$, so $\mathrm{\mid }{a}^{11}\mathrm{\mid }\le M_a=2$, and $\mathrm{Re}(a^{11}\xi \bar{\xi })=a\mathrm{\mid }\xi {\mathrm{\mid }}^2\ge \mathrm{\mid }\xi {\mathrm{\mid }}^2$ for all $\xi \in \mathbb{C}$; hence $L$ is uniformly elliptic with $\theta =1$, $M_a=2$, $M_b=M_c=0$. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

The Heaviside class $H={\mathbf{1}}_{(0,\mathrm{\infty })}$ on $I=(-1,1)$ has no weak derivative in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(I)$: if $v\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(I)$ satisfied the weak-derivative identity ${\int }_{I}H{\phi }^{\mathrm{\prime }}dx=-{\int }_{I}v\phi dx$ for every $\phi \in C_c^{\mathrm{\infty }}(I)$, then the fundamental theorem of calculus would give ${\int }_{I}v\phi dx=\phi (0)$ for every test $\phi$, whereas the shrinking bumps ${\phi }_{ϵ}(x)=\eta (x\mathrm{/}ϵ)$ (with $\eta$ the published smooth bump equal to $1$ on $[-1\mathrm{/}2,1\mathrm{/}2]$ and supported in $(-1,1)$) satisfy ${\phi }_{ϵ}(0)=1$ and $\mid {\int }_{I}v{\phi }_{ϵ}dx\mid \le {\int }_{[-ϵ,ϵ]}\mathrm{\mid }v\mathrm{\mid }dx\to 0$ as $ϵ\downarrow 0$ by absolute continuity of the integral; this contradiction shows that no locally integrable function represents the distributional derivative, which is the Dirac mass at $0$ on $I$. (Weak derivative of a locally integrable function, Absolute continuity of the integral, A smooth bump between concentric Euclidean balls)

The weak derivative is characterized by $\int u{\phi }^{\mathrm{\prime }}=-\int u^{\mathrm{\prime }}\phi$ for every compactly supported smooth test. Integrating the displayed piecewise formula for $u$ by parts on $(-1,0)$ and $(0,1)$ gives $u^{\mathrm{\prime }}=1$ on $(-1,0)$ and $u^{\mathrm{\prime }}=1\mathrm{/}2$ on $(0,1)$, with no point mass because $u$ is continuous at $0$. Weak differentiation is linear in the class: if a class has a weak derivative in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$, every linear combination with constant coefficients has the corresponding linear combination of weak derivatives. (Weak derivative of a locally integrable function)

Assume Countable Choice. If $u\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(-1,1)$, then its first weak derivative $u^{\mathrm{\prime }}$ has a weak derivative in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$, and that weak derivative is the distributional second derivative of $u$. (Weak derivative of a locally integrable function, The notation $H^k$ and the reserved zero-boundary symbol)