论文标题
Monge-Ampère特征值问题的反向迭代
Inverse Iteration for the Monge-Ampère Eigenvalue Problem
论文作者
论文摘要
我们提出了一种迭代方法，基于反复反转具有差异边界条件的Monge-ampère操作员，并在有界的凸域$ω\ subset \ subset \ mathbb {r}^n $上规定了右侧。我们证明，该方法生成的$ u_k $收敛为$ k \ to \ infty $，以解决monge-ampèreeigenvalue问题$$ $ \ begin {cases} \ text {det} d^2u =λ_ \ text {on} \部分ω。 \ end {cases} $$由于此问题的解决方案是唯一的，直至正乘积常数，因此归一化的$ \ hat {u} _K：= \ frac {u_k} {|| || u_k || u_k || _ {l^{l^{\ infty}（\ infty}（ω）}} $ converute the EigeNFuntun funcuttion to unuctut。此外，我们表明$ \ lim \ limits_ {k \ to \ infty} r（u_k）= \ lim \ limits_ {k \ to \ infty} r（\ hat {\ hat {u} _k）=λ__} （-u）\ \ text {det} d^2u} {\int_Ω（-u）^{n+1}}。$$我们的方法收敛于可以明确构建的各种初始选择$ u_0 $，并且不依赖Monge-ampèreeignvalue $ u_envalue $ $ u_0 $。
We present an iterative method based on repeatedly inverting the Monge-Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $Ω\subset \mathbb{R}^n$. We prove that the iterates $u_k$ generated by this method converge as $k \to \infty$ to a solution of the Monge-Ampère eigenvalue problem $$\begin{cases} \text{det} D^2u = λ_{MA} (-u)^n & \quad \text{in } Ω,\\ u = 0 & \quad \text{on } \partial Ω. \end{cases}$$ Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat{u}_k := \frac{u_k}{||u_k||_{L^{\infty}(Ω)}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim\limits_{k \to \infty} R(u_k) = \lim\limits_{k \to \infty} R(\hat{u}_k) = λ_{MA}$, where the Rayleigh quotient $R(u)$ is defined as $$R(u) := \frac{\int_Ω (-u) \ \text{det} D^2u}{\int_Ω (-u)^{n+1}}.$$ Our method converges for a wide class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue $λ_{MA}$.
