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Applicable Analysis
An International Journal
Volume 4, 1974 - Issue 3
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Original Articles

A priori bounds in the cauchy problem for coupled elliptic systems

Pages 207-221 | Published online: 10 May 2007
 

Abstract

By means of the logarithmic convexity of a suitable functional, an a priori inequality is developed for the sum of the squares of the solutions of the following improperly posed Cauchy problem. Consider the coupled elliptic system Lu = aν+f,Lν= bu+g, where L is a uniformly elliptic differential operator, a,b,f and g are bounded integrable functions with |b(x)|≧b0>0 and ν satisfies a stabilizing condition, and where upper bounds for the error in measurement of the Cauchy data on the initial surface are prescribed. From the a priori estimate uniqueness, stability, and pointwise bounds for the solutions u and n are simultaneously deduced. The bounds are improvable by the Ritz technique. Moreover, the method presented here can be extended to the nonlinear system Lu = f(x, ν), Lν =g(x,u)provided g is a suitable form

This work was supported in part by a University of Tennessee Faculty Research Grant and the Office of Naval Research Grant N 00014-67A-0077-0008

This work was supported in part by a University of Tennessee Faculty Research Grant and the Office of Naval Research Grant N 00014-67A-0077-0008

Notes

This work was supported in part by a University of Tennessee Faculty Research Grant and the Office of Naval Research Grant N 00014-67A-0077-0008

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