A priori bounds in the cauchy problem for coupled elliptic systems

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