论文标题
非对称分数运算符的零件集成在半空间
Integration by parts for nonsymmetric fractional-order operators on a halfspace
论文作者
论文摘要
对于强烈的椭圆形分量运算符$ l $的订单$ 2a $（$ 0 <a <a <1 $），我们显示了一个集成的逐个零件公式，用于在模型情况下，在模型情况下，操作员是$ x $与同质符号x $ halfspace $ yspace $ r^n n”。与$（ - δ）^a $相比，新方面是$ l $是非对称的，既有均匀又一个奇怪的部分。因此，它满足了$μ$的转换条件，而通常$μ\ ne a $。 我们提出了一种复杂的方法，它依赖于$ξ_n$在下层或上部复杂半平面中的因素的分解，并使用降级算子与源自Wiener和Hopf的分解原理相结合。这与Dipierro，Ros-Oton，Serra和Valdinoci最近提出的一种真实的计算方法相反。我们的方法允许$μ$在更大的范围内。 另一个新的贡献是（模型）研究，当$μ> 0 $ $时，非均匀的Dirichlet问题的“大”解决方案。 Here we deduce a "halfways Green's formula" for $L$: $$ \int_{R^n_+} Lu\,\bar v\,dx-\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n-1}}γ_0(u/x_n^{μ-1 }）\，{γ_0（\ bar v/x_n^{μ^*}）} \，dx'，$ $当$ u $求解$ l $的非均匀的dirichlet问题，$ v $ solves $ v $ solves solves for $ l^*$; $μ^*= 2a-μ$。最后，当$ u $和$ v $解决非均匀的dirichlet问题时，我们展示了一个完整的格林公式。在这里，Dirichlet和Neumann的$ U $和$ v $ Enter的痕迹，以及边界上的一阶伪差操作员。
For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a<1$) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is $x$-independent with homogeneous symbol, considered on the halfspace $R^n_+$. The new aspect compared to $(-Δ)^a$ is that $L$ is nonsymmetric, having both an even and an odd part. Hence it satisfies a $μ$-transmission condition where generally $μ\ne a$. We present a complex method, relying on a factorization in factors holomorphic in $ξ_n$ in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows $μ$ in a larger range than they consider. Another new contribution is the (model) study of "large" solutions of nonhomogeneous Dirichlet problems when $μ>0$. Here we deduce a "halfways Green's formula" for $L$: $$ \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.
