Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators

Abstract

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega$, when the symbol terms have a certain twisted parity on the normal to $\partial \Omega$. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d(x)=\mathrm{dist}(x,\partial \Omega )$. Examples include fractional Laplacians $(-\Delta {)}^a$ and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on $R^n\setminus \Omega$ reduce to problems supported on $\bar{\Omega }$, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces $F_{p,q}^s$ and $B_{p,q}^s$, including Hölder-Zygmund spaces $B_{\infty ,\infty }^s$. This leads to optimal Hölder estimates, e.g. for Dirichlet solutions of $(-\Delta {)}^{a}u=f\in L_{\infty }(\Omega )$, $u\in d^a{C}^a(\bar{\Omega })$ when $0<a<1$, $a\ne 1/2$ (in $d^a{C}^{a-\epsilon }(\bar{\Omega })$ when $a=1/2$).

Title slightly changed, 34 pages

References in corpus (2)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Spectral results for mixed problems and fractional elliptic operators

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