Regularity Problems for Some Semi-linear Problems

In this article we consider the regularity problems for some partial differential and integral equations (also and specially related to fractals) containing the semilinearity of the following type 1 $$T^ + f(x) = f_ + (x) = \max (0,f(x)).$$ T + is defined on real functions and it is usually calledtruncation operatorIts properties were studied by many authors, see for example [0sw92], [Tri0113], [Zie89]. We will consider this operator in (real) Besov $$B_{pq}^s (\mathbb{R}^n )$$ and Hardy-Sobolev $$H_p^s (\mathbb{R}^n )$$ spaces and we look in these spaces for the best regularity of the solutions of semi-linear equations below. The final statement on the boundedness and Lipschitz continuity of the operatorT + in these spaces can be found in [Tri00] or [Tri0113], §25. We will see that the operatorT + has rather bad continuity properties in the spaces involved. To overcome the difficulties caused by this circumstance one can try to apply bootstrapping arguments using the lifting property of the corresponding (hypo)elliptic operators as we do it in Section 2. But it may happen (as we see in Sections 4 and 5) that there is no space to start from. An efficient tool to obtain the optimal smoothness also in such case is the so-called Q-method recently invented by H. Triebel in [TriOla] (see also [TriOlb], §27). We describe it briefly in Section 3.