The construction of a function f satisfying \(\Delta f\in L^ 1\) and \(\partial _{ij}f\not\in L^ 1\)
Since a couple of decades it is well known that the equation \(\Delta f=g\) for a given continuous g needs not to have a twice continuously differentiable solution f. From those results one easily deduces the following fact: distributional solutions of \(\Delta f=g\) for \(g\in L^{\infty}\) need not belong to \(H^{2,\infty}\). So, the other end of the half-line (1,\(\infty)\) becomes interesting, too. Because for \(1<p<\infty\) we have elliptic regularity: \(\Delta f=g\) for \(g\in L^ p\) implies \(f\in H^{2,p}\). Although many people consider the \(L^ 1\)- question as folklore, no example seems to exist in literature. The author gives now an example of a (radial symmetric) distributional solution of \(\Delta f=g\) for radial symmetric \(g\in L^ 1\) such that all second (distributional) derivatives are not \(L^ 1\).
scientific article; zbMATH DE number 3951085