Integration by parts for nonsymmetric fractionalorder operators on a halfspace
Abstract
For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a<1$) with real kernel, we show an integrationbyparts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is $x$independent with homogeneous symbol, considered on the halfspace $R^n_+$. The new aspect compared to $(\Delta )^a$ is that $L$ is nonsymmetric, having both an even and an odd part. Hence it satisfies a $\mu $transmission condition where generally $\mu \ne a$. We present a complex method, relying on a factorization in factors holomorphic in $\xi_n$ in the lower or upper complex halfplane, using orderreducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, RosOton, Serra and Valdinoci. Our method allows $\mu $ in a larger range than they consider. Another new contribution is the (model) study of "large" solutions of nonhomogeneous Dirichlet problems when $\mu >0$. Here we deduce a "halfways Green's formula" for $L$: $$ \int_{R^n_+} Lu\,\bar v\,dx\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n1}}\gamma_0(u/x_n^{\mu 1 })\,{\gamma_0(\bar v/x_n^{\mu ^*})}\, dx', $$ when $u$ solves a nonhomogeneous Dirichlet problem for $L$, and $v$ solves a homogeneous Dirichlet problem for $L^*$; $\mu ^*=2a\mu $. Finally, we show a full Green's formula, when both $u$ and $v$ solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of $u$ and $v$ enter, as well as a firstorder pseudodifferential operator over the boundary.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.13964
 arXiv:
 arXiv:2012.13964
 Bibcode:
 2020arXiv201213964G
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 35S15;
 47G30;
 35J25;
 60G52
 EPrint:
 The results in this paper are valid for operators satisfying the mutransmission condition. It was overlooked that the main example L does not satisfy that condition in all cases, but only a principal mutransmission condition. The missing cases are now treated in the subsequent paper arXiv:2104.05581