= Solution
The interior <elliptic regularity> assertion for the <Laplace equation> is that a <harmonic function> is <smooth>, in fact <real analytic>, throughout $U$. No boundary regularity of its unspecified boundary values is implied.
First let $u\in C^2(U)$ and $\Delta u=0$. On a ball compactly contained in $U$, differentiating its spherical average and applying the <divergence theorem> expresses that derivative as a constant factor times $\int_{B_r}\Delta u$, which is zero. The spherical average tends to $u$ at the center as $r\to0$. This proves the <mean value property for harmonic functions>.
Choose a radially symmetric <smooth> <mollifier> $\eta_r$, supported in $B_r$, with integral one. By integrating the spherical <mean value property>,
$$
u(x)=\int\eta_r(x-y)u(y)\,dy
$$
whenever $B_r(x)\Subset U$. For fixed $r$ the right side is a <smooth> <convolution>, since all <derivatives> can be placed on $\eta_r$. Thus $u$ is <smooth>. The same argument proves the <Weyl lemma> for a distributionally <harmonic> $u\in L^1_{\mathrm{loc}}$: first mollify $u$, apply the fixed-radius identity, and let the mollification radius tend to zero in distributions to obtain the same smooth representative.
To prove <real analytic> regularity, differentiating the fixed-radius <convolution> gives the <interior derivative estimate for a harmonic function>
$$
|\partial_j v(x)|\leq\frac{C_\ell}{r}\sup_{B_r(x)}|v|
$$
for any <harmonic> $v$. All <derivatives> of $u$ are <harmonic>. On nested balls between $B_R(a)$ and $B_{R/2}(a)$, apply this estimate $k$ times, decreasing the radius by $R/(2k)$ each time. For $|\alpha|=k$,
$$
\sup_{B_{R/2}(a)}|D^\alpha u|\leq\left(\frac{Ck}{R}\right)^k\sup_{B_R(a)}|u|
\leq\left(\frac{Ce}{R}\right)^k k!\sup_{B_R(a)}|u|.
$$
The inequality $k^k\leq e^k k!$ follows by integrating $\log s$ below the sum defining $\log(k!)$. Apply the one-dimensional <Taylor theorem> along each segment, expanding directional <derivatives> by the multinomial formula. The remainder is bounded by $M(A\sum_j|h_j|)^k$, so it tends to zero for sufficiently small $h$. This gives a locally convergent multivariate <Taylor series>. \b[A <harmonic function> is <real analytic> in the interior.]
The usual inhomogeneous <elliptic regularity> statement also follows: if $\Delta u=f$ and $f$ is <smooth>, take a cutoff $\chi$ equal to one near a given point and set $w=\Phi*(\chi f)$, where $\Phi$ is a <fundamental solution of the Laplace equation> with $\Delta\Phi=\delta$. Moving every <derivative> to the compactly supported <smooth> function $\chi f$ shows $w$ is <smooth>. Locally $u-w$ is <harmonic>, so $u$ is <smooth> there too.
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