论文标题
在Lipschitz的连续性和平滑度上，直到双曲线泊松方程解决方案的边界
On Lipschitz continuity and smoothness up to the boundary of solutions of hyperbolic Poisson's equation
论文作者
论文摘要
我们解决了dirichlet问题$ \ left.u \ right | _ {\ mathbb {b}^n} =φ，$ for超纤维泊松方程$Δ_Hu =μ$ $ $ $ $ $ $ $ $ $ $φ\ in l_1（\ partial \ mathbb {b}^n）$和$ $ $ a的条件，使得A级的条件。 接下来，我们为Lipschitz连续的某些双曲线泊松方程的解决方案提供了简短的证明，该方程以前在\ cite {Chenras}中建立。 此外，我们研究了关于双曲线拉普拉斯的一些替代假设，这些假设与Riesz的潜力有关。此外，事实证明，当地的Hölder连续性可以解决某些双曲线泊松方程。 我们表明，如果$ u $在上半空间中是双曲线谐波，则$ \ frac {\ partial u} {\ partial y}（x_0，y）\ to 0，y \ to 0，y \ to y \ to 0^+$，当功能的边界函数$ f $ u $在边界点上是$ u $的$ f $时，在边界点$ u $可区分$ x_0 $。作为推论，我们显示$ c^1（\ overline {\ mathbb {h}^n}）$双曲谐波函数的平滑度，该函数从$ c_c^1（\ mathbb {r}^{n-1}）复制。
We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=φ,$ for hyperbolic Poisson's equation $Δ_h u=μ$ where $φ\in L_1(\partial \mathbb{B}^n)$ and $μ$ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson's equations, previously established at \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz's potential. Also, local Hölder continuity is proved for solution of certain hyperbolic Poisson's equations. We show that, if $u$ is hyperbolic harmonic in the upper half-space, then $\frac{\partial u}{\partial y}(x_0,y)\to 0, y\to 0^+$, when boundary function $f$ of the functions $u$ is differentiable at the boundary point $x_0$. As a corollary, we show $C^1(\overline{\mathbb{H}^n})$ smoothness of a hyperbolic harmonic function, which is reproduced from the $C_c^1(\mathbb{R}^{n-1})$ boundary values.
