By Anton Savin, Boris Sternin

The current monograph is dedicated to the advanced conception of differential equations. now not but a instruction manual, neither an easy selection of articles, the e-book is a primary try to current a roughly distinctive exposition of a tender yet promising department of arithmetic, that's, the advanced conception of partial differential equations. allow us to attempt to describe the framework of this idea. First, basic examples express that ideas of differential equations are, regularly, ramifying analytic services. and, therefore, aren't usual close to issues in their ramification. moment, making an allowance for those very important houses of recommendations, we will attempt to describe the tactic fixing our challenge. without doubt, one has first to contemplate differential equations with consistent coefficients. The equipment fixing such difficulties is famous within the genuine the ory of differential equations: this can be the Fourier transformation. Un thankfully, one of these transformation had now not but been built for complex-analytic services and the authors needed to build via them selves. this modification is, after all, the most important idea of the entire theory.

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**Additional resources for Differential Equations on Complex Manifolds**

**Example text**

Shapiro in their papers [187], [94], [92]. The investigation was carried out under the following natural assumptions: a) The boundary surface. r =an of the domain n is a real analytic hyper- b) The function f(x) is a restriction to the (real) domain n of some entire function on en, we shall denote this function by the same symbol f(x). As was shown by D. Khavinson and H. S. u. = j, u. has zero of order 2 on r c, where r c is the complexification of r. This problem can be investigated (even in a more general case of operators with constant coefficients in the principal part) with the help of the theory of differential equations in complex domain exactly as was explained above.

3. Differential Equations with Constant Coefficients where H(x) is a relative homology class which can be written out explicitly. On this way one can obtain the following global existence theorem for the Cauchy problem for partial differential equations with constant coefficients. " H (- ;x) is a partial differential operator with conNamely, if H= stant coefficients and X is an analytic set subject to conditions 1o and 2° that is not everywhere characteristic with respect to the operator " then one can prove the existence theorem in classes of infinitely H, continuable functions .

2); here we do not distinguish homology classes and contours which represent them. So we must cut the cycle t5h(x) by some shift h'(x) of the cycle h(x) lying on t5h(x). After such a cutting the cycle t5h(x) becomes unclosed. To overcome this difficulty, we use the fact that the cycle we construct is the relative homology, cycle with boundary in X. Thus, we can choose the contour (x) whose boundary consists of h h two parts (one of them is h'(x) and the other lies in X) and clue two copies 1 (x) and h2 (x) with different orientations to the cycle t5h(x) (which is cut along h'(x)), as shown in Fig.