Define the Newtonial potential in a bounded domain by

If is bounded and integrable on , then . And if is bounded and locally holder continuous in then and . But when is only continuous, is not necessarily secondly differentiable. Here is a counterexample.

Let be a homogeneous harmonic polynomial of degree 2. Suppose for some multi-index , for instance . Choose a cut-off function with when . Denote , and let as and divergent. Define

Prove is a continuous function but has no solution near the origin.

If , then . If , then there exists only one such that . Since is a harmonic polynomial and has compact support,

So . As , , so , which means is continuous at 0.

Let , then it is easy to prove is well defined and for and the unique such that ,

(1) means but , otherwise which does not exist.

Suppose there exists is a solution at , (2) means that on . Since is bounded, by the removable singularity theorem, is a harmonic function in , thus an analytic function. However this means is a function on . Contradiction.

Gilbarg Trudinger’s book. Chapter 4. Exercise 4.9

is not bounded in general, because can not be proved bounded directly. But we have . In fact suppose , some , then

Because ,

So as . Then one can use the Removable singularity theorem(Bochner), or the conclusion of 3.7 in GT’s book.