I don't believe it necessary that the dimension of the graph of $\varphi$ be larger than 2. An example is provided by examining Poisson's integral formula for the upper half plane:

$$
  u(x,y) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{y}{(x-\chi)^2+y^2}f(\chi)d\chi.
$$

This function is known to be differentiable off the $x$-axis, continuous on the plane, and satisfy $u(x,0)=f(x)$. The forumula holds as long as $f$ is continuous and
$$
\int_{-\infty}^{\infty} \frac{|f(x)|}{1+x^2} \, dx
$$
converges. In particular, you may take $f$ to be a continuous, bounded function whose graph is known to have Hausdorff dimension larger than 1.