Integration by parts for nonsymmetric fractional-order operators on a halfspace (2012.13964v4)

Abstract: For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a\<1$) with real kernel, we show an integration-by-parts 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 order-reducing 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, Ros-Oton, 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 $L$: $$ \int_{R^n_+} Lu\,\bar v\,dx-\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n-1}}\gamma_0(u/x_n^{\mu -1 })\,{\gamma_0(\bar v/x_n^{\mu ^*})}\, dx', $$ 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 first-order pseudodifferential operator over the boundary.