求解具有指定密度的狄利克雷问题
Solving the Dirichlet problem with prescribed density
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中文总结 AI 辅助
本文针对欧氏空间中具有指定密度的完全非线性椭圆方程狄利克雷问题，证明了其光滑解的存在性，得到最优先验估计，填补了此前仅存在连续解的研究空白。
中文摘要 AI 辅助
本文证明如下结果：设Ω是Rⁿ（n≥3）中的有界、严格凸、光滑区域，φ是从∂Ω到R的光滑函数。则对任意z∈Ω，存在c₁=c₁(Ω, {z}, n, φ)>0，使得当c≥c₁时，问题：在Ω̄\{z}中σ_{n-1}(D²u)=0，u在∂Ω上的限制为φ，且lim_{r→0} sup_{B_r(z)} (u(x)-u(z))/|x-z|^((n-2)/(n-1))=c，在Ω̄\{z}中存在光滑解。此外，我们得到该解的最优先验估计，特别地，对任意整数m≥0，存在C_m=C_m(m, Ω, {z}, n, φ, c)>0，使得|Dᵐu(x)|<C_m|x-z|^((n-2)/(n-1)-m)。本研究首次证明了欧氏空间中具有指定密度的完全非线性椭圆方程狄利克雷问题光滑解的存在性，此前仅得到连续解。
英文摘要
In this paper we prove the following result. Let $Ω\subset\mathbb R^n, n\geq 3,$ be a bounded, strictly convex, smooth domain and $φ: \partialΩ\rightarrow\mathbb R$ be a smooth function. Then for any $z\inΩ,$ there exists $c_1=c_1(Ω, \{z\}, n, φ)>0,$ such that if $c\geq c_1$ then the problem: $σ_{n-1}(D^2 u)=0$ in $\barΩ\setminus\{z\},$ $u|_{\partialΩ}=φ,$ $\lim\limits_{r\rightarrow 0}\sup\limits_{B_r(z)}\frac{u(x)-u(z)}{|x-z|^{(n-2)/(n-1)}}=c,$ admits a smooth solution in $\barΩ\setminus\{z\}.$ Moreover, we obtain the optimal a priori estimates for this solution. In particular, we show for any integer $m\geq 0$ there exists $C_m=C_m(m, Ω, \{z\}, n, φ, c)>0$ such that $|D^m u(x)|<C_m|x-z|^{\frac{n-2}{n-1}-m}.$ This work provides the first result demonstrating the existence of smooth solutions to the Dirichlet problem for fully nonlinear elliptic equations with prescribed density in Euclidean space. Previously, only continuous solutions were obtained.
