Let Ω be a bounded domain of ${\mathbb{R}}^N$ with smooth boundary, $g_1$, $g_2:\mathrm{\Omega }\to \mathbb{R}$ and let $A:D\left(A\right)\to L^2\left(\mathrm{\Omega }\right)$ be a self-adjoint operator defined on a dense subspace $D\left(A\right)\subset L^2\left(\mathrm{\Omega }\right)$ such that A has an orthonormal basis of eigenfunctions in $L^2\left(\mathrm{\Omega }\right)$. For Y >0, giving the function $f:\mathrm{\Omega }\times [0,Y]\to \mathbb{R}$, we consider the problem of finding a function $u:\mathrm{\Omega }\times [0,Y]\to \mathbb{R}$ such that $\begin{array}{rl}& -c\left(y\right)Au(x,y)+u_{yy}(x,y)=f(x,y),x\in \mathrm{\Omega },0<y<Y,\\ & u(x,0)=g_1\left(x\right),u_y(x,0)=g_2\left(x\right),x\in \mathrm{\Omega }.\end{array}$ In the system, the function $c:[0,Y]\to (0,\mathrm{\infty })$ is unknown and approximated by a given function $\mu :[0,Y]\to (0,\mathrm{\infty })$ such that $\underset{0⩽y⩽Y}{sup}\left\{\left|c\left(y\right)-\mu \left(y\right)\right|+\left|\frac{\mathrm{d}c}{\mathrm{d}y}\left(y\right)-\frac{\mathrm{d}\mu }{\mathrm{d}y}\left(y\right)\right|+\left|\frac{{\mathrm{d}}^{2}c}{\mathrm{d}{y}^2}\left(y\right)-\frac{{\mathrm{d}}^2\mu }{\mathrm{d}{y}^2}\left(y\right)\right|\right\}⩽\delta$ for a $\delta >0$ small. This Cauchy problem is ill-posed and has many applications in physics and other fields. For this reason, a regularization for the problem is in order.

中文翻译：

---

非常数系数椭圆方程的一个非齐次柯西问题

令Ω是RN的有界域，边界光滑，g 1 , g 2 : Ω → R 并令 A : D ( A ) → L 2 ( Ω ) 是定义在稠密子空间 D ( A ) 上的自伴随算子⊂ L 2 ( Ω ) 使得 A 有...