By Thomas H. Otway (auth.)
Partial differential equations of combined elliptic-hyperbolic variety come up in different components of physics and geometry, together with fluid and plasma dynamics, optics, cosmology, site visitors engineering, projective geometry, geometric variational thought, and the speculation of isometric embeddings. And but even the linear concept of those equations is at a really early level. this article examines a number of Dirichlet difficulties which are formulated for equations of Keldysh style, one of many major periods of linear elliptic-hyperbolic equations. Open boundary stipulations (in which info are prescribed on merely a part of the boundary) and closed boundary stipulations (in which facts are prescribed at the complete boundary) are either thought of. Emphasis is at the formula of boundary stipulations for which ideas will be proven to exist in a suitable functionality house. particular functions to plasma physics, optics, and research on projective areas are mentioned. (From the preface)
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24) implies that ux D 0 on L2 : We use this last identity as Cauchy data for the Cauchy–Kowalevsky Theorem, to argue that u remains equal to zero as one moves in the negative x-direction away from L2 along the rectangle D C : This argument was applied in . 5). 21) is suitably interpreted. 1. 1. Proof. 1. 21) is uniquely determined by data given on the non-characteristic boundary. So the problem is over-determined for sufficiently smooth solutions if data are given on the entire boundary. This completes the proof.
0; b/ D 0; that constant is zero. Thus on the rectangle @D C we have a closed Dirichlet problem having homogeneous boundary conditions. 1 of Sect. 1 implies that the C 2 function u attains both its maximum and minimum values on the boundary. Because it is identically zero there, u must be zero in all of D C : We obtain the identical vanishing of u on the hyperbolic region by integration along characteristic lines as in . 30) Thus is nonincreasing in y on any arbitrarily chosen characteristic.
27) y Ä 0 on that vertical line. 28) are in contradiction unless c1 D c2 : Taking into account that cannot increase with increasing y on the line x D 0; it also cannot decrease with increasing y; as it would then have to increase in order to return to its initial value at the endpoint. This implies that y D 0 on the y-axis. 0; y/ is constant there. 0; b/ D 0; that constant is zero. Thus on the rectangle @D C we have a closed Dirichlet problem having homogeneous boundary conditions. 1 of Sect. 1 implies that the C 2 function u attains both its maximum and minimum values on the boundary.