Generalized Solutions of a Periodic Goursat Problem

Under reasonable assumptions on the data u, v and the function f , we show that the nonlinear periodic Goursat problem ∂u ∂x∂y (x, y) = f(x, y, u(x, y)); u(x, 0) = v(x); u(0, y) = w(y) which cannot be posed in the general theory of distributions, may be studied and solved in a differential algebra of periodic new generalized functions on R2. This algebra contains, in a canonical way, the space of periodic distributions on R2 as a linear subspace. When the data are Dirac measures, a particular study is done, showing the nature of the singularities involved in this case. 1991 Mathematics Subject Classification. 46F10, 42B05, 35D99.