Abstract
In these notes we consider second order linear elliptic boundary
value problems in the framework of different spaces on continuous
functions. We appeal to a general formulation which contains some
interesting particular cases as, for instance, a new class of
functional spaces, called here Hölog spaces and denoted by the
symbol $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,,$ $\,0 \leq\,\lambda<\,1\,,$ and
$\,\alpha \in\,\mathbb{R}\,.$ One has the following inclusions
$$
C^{0,\,\lambda+\,\epsilon}(\overline{\Omega})\subset \,C^{0,\,\lambda}_\alpha(\overline{\Omega})\subset
\,C^{0,\,\lambda}(\overline{\Omega}) \subset \,C^{0,\,\lambda,}_{-\alpha}(\overline{\Omega})
\subset\,C^{0,\,\lambda-\,\epsilon}(\overline{\Omega})\,,
$$
for $\,\alpha>\,0\,$ ($\epsilon >\,0\,$ arbitrarily small). Roughly
speaking, for each fixed $\,\lambda\,,$ the family $\,
C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ is a refinement of the single Hölder
classical space $\, C^{0,\,\lambda}(\overline{\Omega})=\,C^{0,\,\lambda}_0(\overline{\Omega})\,.$ On the
other hand, for $\,\lambda=\,0\,$ and $\,\alpha>\,0\,,$
$\,C^{0,\,0}_\alpha(\overline{\Omega})=\,\, D^{0,\,\alpha}(\overline{\Omega})\,$ is a Log space. The
more interesting feature is that, as for classical Hölder (and
Sobolev) spaces, full regularity occurs. namely, for each
$\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u $ and
$\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ regularity.
All the above setup is presented as part of a more general
picture.
Mathematics Subject Classification: Primary: 31B10; Secondary: 31B35, 33E30, 35A09, 35B65, 35J25, 58F15, 58F17.
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