The Hölder space is thus the space of the -time continuously differentiable, limited functions from after , whose -th partial derivatives are Hölder-continuous to a constant and also limited. In special cases , one usually speaks of Lipschitz continuity .
Kellogg's theorem
Let and be a bounded domain with - edge and a strictly elliptic operator in with coefficients in , ie
,
where in lie and the matrix is the ellipticity condition
for all
satisfied with one of independent constants . Let the function be non-positive and and . Then has the equation
a clear classic solution .
Since the above equation has no classical solution , if only continuity is required, the control of the continuity module is relevant for existence theory in the theory of partial differential equations. Holder spaces are a class of functions within which classical existential theory can be operated.
literature
HW Alt: Linear functional analysis. 4th edition, Springer-Verlag, ISBN 3-540-43947-1 .
D. Gilbarg, NS Trudinger: Elliptic Partial Differential Equations of Second Order. In: Basic Teachings of the Mathematical Sciences. Volume 224, Springer-Verlag, Berlin / Heidelberg / New York 1977, ISBN 3-540-08007-4 .