Invertibility of some heat potentials in BMO norms
For \(C^ 1\) domains D in \(R^ n\) and \(L^ p\) boundary data the initial-Dirichlet problem \(\Delta_ xu-D_ t=0\) in the cylinder \(D\times (0,T),\lim_{t\to 0} u(X,t)=0\) uniformly on compacts in D, u(X,t)\(\to f(P,s)\) on the lateral surface \(\partial D\times (0,T)\), is analyzed. Continuity of the boundary integral \([Jf](P,t)=\lim_{\epsilon \to 0} \iint^{t-\epsilon}_{\partial D}K(P,Q,t-s)dQ ds\)where K is the corresponding kernel of the double-layer heat potential, was proved in the two first authors' previous work [ibid. 72, 289-305 (1984; Zbl 0561.35037)]. In the present paper the invertibility of the boundary terms \(cI+J\), \(c\neq 0\) and I-identity operator, in \(B_ 0 MOC(\partial D\times (0,T))\) is established, where \(B_ 0 MOC\) is a subspace of functions f with caloric bounded mean oscillation (BMOC) having bounded initial behaviour at \(t=0\).
scientific article; zbMATH DE number 3996199