A non-homogeneous cauchy problem for an elliptic equation with non-constant coefficient

Abstract

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.

COMMUNICATED BY:

• J. Sun

Keywords:

• Ill-posed problem
• cauchy problem
• semi-linear elliptic equation
• fourier truncated regularization method

2010 Mathematics Subject Classifications:

• 35J61
• 45D05
• 65J20
• 65R30

Acknowledgments

The authors would like to thank the anonymous referee for constructive criticisms leading to the improvement of our paper. The paper is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.321.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The paper is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.02-2019.321].