First Order Elliptic Systems: A Function Theoretic Approach by Robert P. Gilbert

By Robert P. Gilbert

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Hence DJ(llu = u in the Sobolev sense. To justify the use of Fubini’s theorem we note that s MyU,81, for some M and MI. 23 If w E LiOc(@),u E L’(8), and u = Dw in @, then w = @ Jmu, where @ is hyperanalytic. Conversely, if@is hyperanalytic and u E L1(@),then @ + Jmu E L;oc(@)and D(@ + JCVU)= u in 8. Thus @ is hyperanalytic. D(@ + Jcru) = 0 is obvious. ) will denote a constant that depends upon whatever constants, functions, or domains are listed within the parentheses. Also q will continue to denote the hypercomplex function arising from the original system of differential equations.

3,= Re The kernels rl and IN THE PLANE IIc ( u , z ) A - ' dug. 53) 51 5. CAUCHY REPRESENTATION Thus we have (T - z ) r j ( z ,T ) E L2,0(C)x L2(C)f o r j = 1 , 2. Note that L2v2(C)= L2(C). 52) determines r,(z, T ) as a function of z with T as a parameter. 53) determines r j ( z , T ) as a function of 7. Let us summarize in a theorem. 49). 50) and (1 A), respectively, where the resolvent kernels are uniquely determined by Eqs. 53). Also let us note the following regularity result. 44 Let h E CoTP(C) for some 0 < P < 1 .

Thus k o w = h has a unique solution for each h E L2(C,). This solution 47 5. 44) into $tow = h and noting that the resulting equation must be true for all h E L2(C0),we obtain the integral equations for r, Also, the identity w = R[fiow]must be satisfied for all w E L*(C,). This yields the two additional integral equations rr 2 '{W'CZ), 2 /=I w1(7)} = 0. 46) 48 1. ELLIPTIC SYSTEMS IN THE PLANE In obtaining these relations we used w' = R[u'], which is a consequence of Elow' = v'. If solvability condition [h, v'] = 0 is met, then a solution of &low = h is also a solution of Mow = h since + [w, w'l 0 = [fiow, v'l = [Mow, v'l = [w,Mo*v'] + [w, w'] = [w, w'l, which implies that f i o w = Mow.

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