论文标题
关于通过解决最小表面方程的外部迪里奇特问题的解决方案存在叶子的存在
On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation
论文作者
论文摘要
给定一个外部域$ω$，带有$ c^{2，α} $边界，$ \ mathbb {r}^{n} $，$ n \ geq3 $，我们获得了$ 1 $ -Parameter $ -Parameter $_γ\ in C^{\ infty}最小表面方程的解决方案的\leqπ/2 $，使得，如果$ \ weft \ welet \vertγ\ right \ vert \ vert <π/2 $，$u_γ\ in C^{\ infty} \ in c^{\ infty} \ left（ω\ weft（ω\ right） $u_γ| _ {\partialΩ} = 0 $，$ \ max _ {\partialΩ} \ left \ nabla \ nabla \ nabla \ nabla \ nablau_γ\ right \ right \ vert = \tanγ$，如果$ \ \ left \ left \ left \ welet \vertγ\ right \ right \ right \ vert = vert = pert = per} $m_γ\ subset \overlineΩ\ times \ times \ mathbb {r} $带有$ \ partialm_γ= \partialΩ$。这些函数中的每一个都是界限的，渐近\ [c_γ= \ lim _ {\ left \ vert x \ right \ right \ vert \ rightarrow \ rightarrow \ infty}u_γ\ left（x \ right）。 \]映射$γ\ rightArrowu_γ\ left（x \ right）$（对于固定的$ x \inΩ$）和$γ\ rightarrowc_γ$严格增加和界限。这些函数的图形折叠了$ \ mathbb {r}^{n+1} $ \ [\ left \ {\ left（x，x，z \ right）\inΩ\ times \ mathbb {r} x \ right）\ right \}。此外，如果$ \ mathbb {r}^{n} \backslashΩ$满足最大半径$ρ$的内部球体条件，并且如果$ \partialΩ$包含在最小半径$ \ varrho $的球中0，c_ {π/2} \ right] \ subset \ left [0，σ_{n} \ varrho \ right]，\]，其中\ [σ_{n} = \ int_ {1}}^{1}^{\ infty} {\ frac} \ frac {dt} } -1}}。 \]上述夹杂物之一是当且仅当$ρ= \ varrho $，$ω$的外部是半径$ρ$的外部，而解决方案是径向的。
Given an exterior domain $Ω$ with $C^{2,α}$ boundary in $\mathbb{R}^{n}$, $n\geq3$, we obtain a $1$-parameter family $u_γ\in C^{\infty}\left(Ω\right) $, $\left\vert γ\right\vert \leqπ/2$, of solutions of the minimal surface equation such that, if $\left\vert γ\right\vert <π/2$, $u_γ\in C^{\infty}\left( Ω\right) \cap C^{2,α}\left( \overlineΩ\right) $, $u_γ|_{\partialΩ}=0$ with $\max_{\partialΩ}\left\Vert \nabla u_γ\right\Vert =\tanγ$ and, if $\left\vert γ\right\vert =π/2$, the graph of $u_γ$ is contained in a $C^{1,1}$ manifold $M_γ\subset\overlineΩ\times\mathbb{R}$ with $\partial M_γ=\partialΩ$. Each of these functions is bounded and asymptotic to a constant \[ c_γ=\lim_{\left\Vert x\right\Vert \rightarrow\infty}u_γ\left( x\right) . \] The mappings $γ\rightarrow u_γ\left( x\right) $ (for fixed $x\inΩ$) and $γ\rightarrow c_γ$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of $\mathbb{R}^{n+1}$ \[ \left\{ \left( x,z\right) \inΩ\times\mathbb{R}\text{, }-u_{π/2}\left( x\right) <z<u_{π/2}\left( x\right) \right\} . \] Moreover, if $\mathbb{R}^{n}\backslashΩ$ satisfies the interior sphere condition of maximal radius $ρ$ and if $\partialΩ$ is contained in a ball of minimal radius $\varrho$, then \[ \left[ 0,σ_{n}ρ\right] \subset\left[ 0,c_{π/2}\right] \subset\left[ 0,σ_{n}\varrho\right] , \] where \[ σ_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}. \] One of the above inclusions is an equality if and only if $ρ=\varrho$, $Ω$ is the exterior of a ball of radius $ρ$ and the solutions are radial.
