My purpose is to derive some regularity result concerning the following integral equation

where is open and bounded and . To this purpose, in this entry we first consider the equation

for a suitable choice of .

**The case **. We will prove that for any . Indeed, up to a constant factor, the first derivative of are given by

.

From this formula,

.

By the intermediate value theorem, on the line from to , there exists some with

.

We put

.

Since is bounded, we can find with and replace the integral on by the integral on . We now write

Clearly, by co-area formula

Besides,

Hence if , we get

.

Obviously, for each fixed, we can find a such that

.

In other words, .

**The case **. We will prove that for any . Up to a constant factor, the second derivatives of are given by

.

For simplicity, we denote

which is nothing but

.

Obviously

.

We now write

.

As before, we get the existence of with the following property

.

We again put

and write

.

The last integral can be spitted into two parts

.

Obviously

.

Moreover,

and the first integral vanishes. An estimate on will give us

due to the domain of . Thus

.

Hence there is a satisfying

.

The proof now follows.

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