Goursat’s problem with boundary functions in \(L_ 2\). (English. Russian original) Zbl 0672.35041

Kapustin, N. Yu.

Differ. Equations 23, No. 7, 820-828 (1987); translation from Differ. Uravn. 23, No. 7, 1219-1231 (1987).

The author obtains an a priori estimate of a solution u of a Goursat problem, i.e., solving the equation \[ u_{\xi \eta}+a(\xi,\eta)u_{\xi}+b(\xi,\eta)u_{\eta}+c(\xi,\eta)u=f(\xi,\eta) \] on a rectangle ABCD with given (square integrable

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) values of u on the sides AB and AD. This estimate is established in the class of square integrable functions

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whose derivatives \(u_{\xi}\), \(u_{\eta}\) and \(u_{\xi \eta}\) belong to \(L_ 2\) and such that \(u(A)=0\). The estimate asserts that a corresponding norm of u is bounded above by \(L_ 2\)-norms of u and boundary values of u on AB and AD. Then he shows that without the condition \(u(A)=0\) the estimate is not valid. The second part of the present article is devoted to a discrete analogy of the above estimate, i.e. for a solution of the corresponding differential problem.

Reviewer: L.Lebedev

MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
35B45	A priori estimates in context of PDEs
65Z05	Applications to the sciences

Keywords:

a priori estimate; Goursat problem; square integrable functions; differential problem