Integration by parts for nonsymmetric fractional-order operators on a halfspace
英文摘要：
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 $(-Δ)^a$ is that $L$ is nonsymmetric, having both an even and an odd part. Hence it satisfies a $μ$-transmission condition where generally $μ\ne a$. We present a complex method, relying on a factorization in factors holomorphic in $ξ_n$ 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 $μ$ in a larger range than they consider. Another new contribution is the (model) study of "large" solutions of nonhomogeneous Dirichlet problems when $μ>0$. Here we deduce a "halfways Green's formula" for $L$: $$ \int_{R^n_+} Lu\,\bar v\,dx-\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n-1}}γ_0(u/x_n^{μ-1 })\,{γ_0(\bar v/x_n^{μ^*})}\, dx', $$ when $u$ solves a nonhomogeneous Dirichlet problem for $L$, and $v$ solves a homogeneous Dirichlet problem for $L^*$; $μ^*=2a-μ$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
