An inverse problem for the equation $▵u=-cu-d$

Vogelius, Michael. "An inverse problem for the equation $\triangle u=-cu-d$." Annales de l'institut Fourier 44.4 (1994): 1181-1209. <http://eudml.org/doc/75092>.

@article{Vogelius1994,
abstract = {Let $\Omega $ be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem\begin\{\}\triangle u = -cu-d \ \text\{in\} \Omega ,\quad u=0 \ \text\{on\} \partial \Omega .\end\{\}We prove two results: 1) If $\Omega $ is not a ball and if one considers only solutions with $-cu-d\le 0$, then there exist at most finitely many pairs of coefficients $(c,d)$ so that the normal derivative $\{\{\partial u\} \over \{\partial \nu \}\}\vert _\{ \partial \Omega \}$ equals a given $\psi \ne 0$.2) If one imposes no sign condition on the solutions but one additionally supposes that $\Omega $ is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients $(c,d)$ so that $\{\{\partial u\} \over \{\partial \nu \}\}\vert _\{ \partial \Omega \}$ equals a given non-degenerate $\psi $. Our analysis is related to work on the Pompeiu–Schiffer conjectures. To illustrate this relation we also show how our analysis provides a very elementary and short proof of a result, due to Berenstein, concerning the Schiffer conjecture.},
author = {Vogelius, Michael},
journal = {Annales de l'institut Fourier},
keywords = {Radon transform; stationary phase; Schiffer conjecture},
language = {eng},
number = {4},
pages = {1181-1209},
publisher = {Association des Annales de l'Institut Fourier},
title = {An inverse problem for the equation $\triangle u=-cu-d$},
url = {http://eudml.org/doc/75092},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Vogelius, Michael
TI - An inverse problem for the equation $\triangle u=-cu-d$
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 1181
EP - 1209
AB - Let $\Omega $ be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem\begin{}\triangle u = -cu-d \ \text{in} \Omega ,\quad u=0 \ \text{on} \partial \Omega .\end{}We prove two results: 1) If $\Omega $ is not a ball and if one considers only solutions with $-cu-d\le 0$, then there exist at most finitely many pairs of coefficients $(c,d)$ so that the normal derivative ${{\partial u} \over {\partial \nu }}\vert _{ \partial \Omega }$ equals a given $\psi \ne 0$.2) If one imposes no sign condition on the solutions but one additionally supposes that $\Omega $ is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients $(c,d)$ so that ${{\partial u} \over {\partial \nu }}\vert _{ \partial \Omega }$ equals a given non-degenerate $\psi $. Our analysis is related to work on the Pompeiu–Schiffer conjectures. To illustrate this relation we also show how our analysis provides a very elementary and short proof of a result, due to Berenstein, concerning the Schiffer conjecture.
LA - eng
KW - Radon transform; stationary phase; Schiffer conjecture
UR - http://eudml.org/doc/75092
ER -