PUB - Publikationen an der Universität Bielefeld

In this paper, we establish Schauder's estimates for the following non-local equations in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>{d}$$\end{document} : partial differential tu=L kappa,sigma(alpha)u+b center dot backward difference u+f,u(0)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _t u=\mathscr {L}<^>{(\alpha )}_{\kappa ,\sigma } u+b\cdot \nabla u+f,\ u(0)=0, $$\end{document}where alpha is an element of(1/2,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1/2,2)$$\end{document} and b:R+xRd -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ b:\mathbb {R}_+\times \mathbb {R}<^>d\rightarrow \mathbb R$$\end{document} is an unbounded local beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-order Holder function in x uniformly in t, and L kappa,sigma(alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {L}<^>{(\alpha )}_{\kappa ,\sigma }$$\end{document} is a non-local alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable-like operator with form: L kappa,sigma(alpha)u(t,x):=integral Rd(u(t,x+sigma(t,x)z)-u(t,x)-sigma(t,x)z(alpha)center dot backward difference u(t,x))kappa(t,x,z)nu(alpha)(dz),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathscr {L}<^>{(\alpha )}_{\kappa ,\sigma }u(t,x):=\int _\mathbb {R}<^>d\Big (u(t,x+\sigma (t,x)z)-u(t,x)-\sigma (t,x)z<^>{(\alpha )}\cdot \nabla u(t,x)\Big )\kappa (t,x,z)\nu <^>{(\alpha )}(\mathord {\textrm{d}} z), $$\end{document}where z(alpha)=z1 alpha is an element of(1,2)+z1|z|<= 11 alpha=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z<^>{(\alpha )}=z\textbf{1}_{\alpha \in (1,2)}+z\textbf{1}_{|z|\le 1}\textbf{1}_{\alpha =1}$$\end{document}, kappa:R+xR2d -> R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69 pt} \begin{document}$$\kappa :\mathbb {R}_+\times \mathbb {R}<^>{2d}\rightarrow \mathbb {R}_+$$\end{document} is bounded from above and below, sigma:R+xRd -> Rd circle times Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma :\mathbb {R}_+\times \mathbb {R}<^>{d}\rightarrow \mathbb {R}<^>d\otimes \mathbb {R}<^>d$$\end{document} is a gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}-order Holder continuous function in x uniformly in t, and nu alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu <^>{\alpha )}$$\end{document} is a singular non-degenerate alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable Levy measure.