Holder room

In mathematics , the Hölder space (after Otto Hölder ) is a Banach space of functions that plays a role in the theory of partial differential equations . There, Hölder rooms are a natural choice for practicing existential theory.

definition

Be . The Hölder space is the set of all functions with , for which the following norm is finite: ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle C^{k,\gamma }(U)}$ ${\displaystyle u\colon U\rightarrow \mathbb {R} }$ ${\displaystyle u\in C^{k}(U)}$

${\displaystyle \|u\|_{C^{k,\gamma }(U)}:=\sum _{|\alpha |\leq k}\|D^{\alpha }u\|_{C(U)}+\sum _{|\alpha |=k}[D^{\alpha }u]_{C^{0,\gamma }(U)}}$.

Marked here

${\displaystyle \|D^{\alpha }u\|_{C(U)}:=\sup \left\{|D^{\alpha }u(x)|\ |\ x\in U\right\}}$

the supreme norm and

${\displaystyle [D^{\alpha }u]_{C^{0,\gamma }(U)}:=\sup \left\{\left.{\frac {|D^{\alpha }u(x)-D^{\alpha }u(y)|}{|x-y|^{\gamma }}}\ \right|\ x,y\in U,x\neq y\right\}}$

a semi-norm . For one writes . ${\displaystyle C^{0,\gamma }(\Omega )}$ ${\displaystyle C^{\gamma }(\Omega )}$

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 . ${\displaystyle k}$ ${\displaystyle U}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle k}$ ${\displaystyle \gamma \in (0,1]}$ ${\displaystyle \gamma =1}$

Kellogg's theorem

Let and be a bounded domain with - edge and a strictly elliptic operator in with coefficients in , ie ${\displaystyle \gamma \in (0,1]}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle C^{2,\gamma }}$ ${\displaystyle L}$ ${\displaystyle \Omega }$ ${\displaystyle C^{\gamma }(\Omega )}$

${\displaystyle Lu:=\sum _{i,j=1}^{n}a^{ij}(x)\cdot (D^{ij}u)(x)+\sum _{i=1}^{n}b^{i}(x)\cdot (D^{i}u)(x)+c(x)\cdot u(x)}$,

where in lie and the matrix is the ellipticity condition ${\displaystyle a^{ij},b^{i},c:\Omega \rightarrow \mathbb {R} }$ ${\displaystyle C^{\gamma }(\Omega )}$ ${\displaystyle A(x):=(a^{ij}(x))_{i,j=1,\ldots ,n}}$

${\displaystyle \langle A(x)\xi ,\xi \rangle \geq \lambda \|\xi \|^{2}}$ for all ${\displaystyle x\in \Omega ,\xi \in \mathbb {R} ^{n}}$

satisfied with one of independent constants . Let the function be non-positive and and . Then has the equation ${\displaystyle x,\xi }$ ${\displaystyle \lambda >0}$ ${\displaystyle c\leq 0}$ ${\displaystyle f\in C^{\gamma }(\Omega )}$ ${\displaystyle \varphi \in C({\overline {\Omega }})\cap C^{2,\gamma }(\Omega )}$

${\displaystyle \left\{{\begin{array}{rlll}Lu&=&f&{\textrm {in}}\ \Omega \ ,\\u&=&\varphi &{\textrm {auf}}\ \partial \Omega \ ,\end{array}}\right.}$

a clear classic solution . ${\displaystyle u\in C({\overline {\Omega }})\cap C^{2,\gamma }(\Omega )}$

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. ${\displaystyle u}$ ${\displaystyle f}$

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 .