Measurable coefficients with a Holder-regular weak solution

Given: The Axiom of Choice and Countable Choice; the annulus $\mathrm{\Omega }=\{x\in {\mathbb{R}}^2:\frac{1}{4}<\mathrm{\mid }x\mathrm{\mid }<1\}$; the radial coefficient $a$ equal to $1$ for $\mathrm{\mid }x\mathrm{\mid }<\frac{1}{2}$ and $4$ for $\mathrm{\mid }x\mathrm{\mid }>\frac{1}{2}$; and the radial function $u$ defined by the two displayed formulas.

Uniform ellipticity and boundedness: $a$ is measurable, $1\le a\le 4$ on $\mathrm{\Omega }$, and the matrix $A=a\mathrm{I}\mathrm{d}$ satisfies $\mathrm{\mid }\xi {\mathrm{\mid }}^2\le ⟨A(x)\xi ,\xi ⟩\le 16\mathrm{\mid }\xi {\mathrm{\mid }}^2$ for all $\xi \in {\mathbb{R}}^2$, so the ellipticity constant is $\theta =1$ and the coefficient bound is $M_a=4$ in the convention of Uniformly elliptic divergence-form operators and their sesquilinear forms (The notation $H^k$ and the reserved zero-boundary symbol).

Regularity of the pieces: on each of the open annuli $U_1=\{\frac{1}{4}<\mathrm{\mid }x\mathrm{\mid }<\frac{1}{2}\}$ and $U_2=\{\frac{1}{2}<\mathrm{\mid }x\mathrm{\mid }<1\}$ the function $u$ is smooth and radial, with $\mathrm{\nabla }u=\frac{1}{r}\frac{\mathrm{\partial }u}{\mathrm{\partial }r}x$ and ${\mathrm{\partial }}_{r}u=1\mathrm{/}r$ on $U_1$, ${\mathrm{\partial }}_{r}u=1\mathrm{/}(4r)$ on $U_2$; the glued function lies in $H^1(\mathrm{\Omega })$ with these a.e. gradients. Indeed $u=G(\mathrm{\mid }x\mathrm{\mid })$, where $G(t)=-\log 4$ for $t\le 1\mathrm{/}4$, $G(t)=\log t$ for $1\mathrm{/}4<t\le 1\mathrm{/}2$, and $G(t)=\frac{1}{4}\log t-\frac{3}{4}\log 2$ for $t>1\mathrm{/}2$. This is a globally $4$-Lipschitz scalar function. Since $x\mapsto \mathrm{\mid }x\mathrm{\mid }$ is smooth on $\bar{\mathrm{\Omega }}$ with gradient $x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ and belongs to $H^1(\mathrm{\Omega })$, the Sobolev chain rule establishes the asserted membership and gradient (Chain rule for globally Lipschitz scalar maps of Sobolev functions) (Weak derivative of a locally integrable function, Holder's inequality for integrals, including the endpoint cases).

Continuity and differentiability across the interface: at $\mathrm{\mid }x\mathrm{\mid }=\frac{1}{2}$ the first formula gives $\log \frac{1}{2}=-\log 2$ and the second gives $\frac{1}{4}\log \frac{1}{2}-\frac{3}{4}\log 2=-\log 2$, so $u$ is continuous there; the radial derivative is $2$ from the inner side and $\frac{1}{2}$ from the outer side, so the derivative is discontinuous and $u\mathrm{\notin }{C}^1(\mathrm{\Omega })$, while $\mathrm{\Omega }$ is bounded away from the origin in polar coordinates, so $u$ is Lipschitz and hence Holder of every exponent $\alpha <1$ (Local Hölder and scaled C-two-alpha norms on balls).

The flux: $a\mathrm{\nabla }u=x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ a.e. on $\mathrm{\Omega }$, because $a(r){\mathrm{\partial }}_{r}u(r)=1\mathrm{/}r$ for both branches of $a$ and of $u$; the field $x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ is smooth on $\mathrm{\Omega }$, $\mathrm{div}(x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2)=0$ there, and $x\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2=\mathrm{\nabla }\log \mathrm{\mid }x\mathrm{\mid }$ (Local weak solutions of a divergence-form operator).

Local weak solutions: $u\in H^1(\mathrm{\Omega })$ is a local weak solution of $-\mathrm{div}(a\mathrm{\nabla }u)=0$ when ${\int }_{\mathrm{\Omega }}a(x)\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx=0$ for every $v\in H_0^1(\mathrm{\Omega })$; the De Giorgi-Nash theorem gives, for such a solution with measurable uniformly elliptic coefficients, a Holder representative with exponent depending only on $n,\theta ,M_a$ (Local weak solutions of a divergence-form operator, De Giorgi-Nash interior Holder regularity for divergence-form equations).