关于具有受限大边界数据的k-黑塞方程逆源问题的仿射截面层析成像
Affine section tomography for inverse source problems in $k$-Hessian equations with restricted large boundary data
AI总结：
研究k-黑塞方程逆源问题，利用非线性狄氏-诺伊曼映射，通过受限大边界数据射线确定正光滑源，给出重建公式和唯一性证明，涵盖不同k值情况。
摘要：
我们考虑在\(\mathbb{R}^n\)中光滑一致凸域的k-容许分支上的k-黑塞方程\(\sigma_k(D^2u)=f(x)\)的逆源问题，其中\(2\leq k\leq n\)。证明非线性狄氏-诺伊曼映射从其在受限大数据射线\(t\phi_E|_{\partial\Omega}\)上的值确定正光滑源，给出重建公式及唯一性，不同k值有不同结果。
英文摘要：
We consider an inverse source problem for the $k$-Hessian equation \begin{equation*} σ_k(D^2u)=f(x) \end{equation*} on the $k$-admissible branch in a smooth uniformly convex domain in $\mathbb R^n$, where $2\le k\le n$. We prove that the nonlinear Dirichlet-to-Neumann map determines the positive smooth source from its values on the restricted large-data rays $tϕ_E|_{\partialΩ}$, where $E\in Gr(k-1,n)$, $ϕ_E(x)=|P_Ex|^2/2$, and $t$ is sufficiently large. The Hessian of each boundary profile has exactly $k-1$ large directions and is flat on $V=E^\perp$. Thus, the leading profile lies on a rank $k-1$ face of the $k$-Hessian structure, and the first source-dependent correction is governed by the missing directions. More precisely, this correction solves fiberwise Poisson equations on the affine sections $Ω\cap(y+V)$ of dimension $q=n-k+1$. We prove that these sectionwise solutions patch smoothly through glancing points where the sections collapse, and we obtain boundary normal derivative asymptotics by local barriers. The boundary flux of the correction gives the section integrals $\int_{Ω\cap(y+V)}f\,d\mathcal H^q$. Varying $E$ yields the affine $q$-plane Radon transform of the zero extension of $f$. We give an explicit reconstruction formula through the Fourier slice identity, and the injectivity of the affine Radon transform gives uniqueness. The endpoint $k=n$ recovers the Monge--Ampère chord/X-ray geometry, while the range $n\ge3$ and $2\le k<n$ gives inverse source results for genuinely non-determinant Hessian equations.
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