On the global solubility of the Monge-Ampere hyperbolic equations

Numerical methods for the hyperbolic Monge-Ampère equation based on the method of characteristics

“…We define C 2 to be an integral strip if there exists a C 1 -strip which can be extended to the C 2 -strip uniquely, solely using the PDE (3) and the strip conditions (9). In this case C 1 is called a free strip.…”

“…Furthermore, we supplement p = u x and q = u y to obtain a first order strip C 1 . If by using the PDE (3) and the strip conditions (9) we are able to determine r = u x x , s = u xy and t = u yy uniquely, then the strip C 2 = {(x(λ), y(λ), u(λ), p(λ), q(λ), r (λ), s(λ), t(λ)) | λ ∈ I } is called an integral strip, and C 1 is a free strip, otherwise C 1 is called a characteristic strip. Note that along a free strip, but not along a characteristic strip, the derivatives u x x , u xy and u yy can all be determined along the strip, either by being interior derivatives with respect to C 1 , or by combining the PDE (3) with the remaining interior derivatives.…”

“…On the other hand, the hyperbolic equation is more exotic and the literature is scarce. Moreover, the most important results regarding existence and uniqueness [8,9] consider Cauchy boundary conditions, while for optical design we require the so-called transport boundary conditions, sometimes also called oblique boundary conditions or second boundary conditions [10], stating that the boundary of a source domain is mapped to the boundary of the target domain by the optical map. The papers [11,12] are the most notable references in the context of numerical results in illumination optics using the hyperbolic Monge-Ampère equation.…”

Differential invariants of generic hyperbolic Monge-Ampère equations

“…In spite of more than 200 years of history of Monge-Ampère equations and numerous publications devoted to them it would be an exaggeration to say that their nature is well understood. An important success was establishing the existence and uniqueness theorems by Lewy and others (see [10,3] for local aspects and [20] for global ones). The classical Monge integration method was modernized by Matsuda [15,16] and Morimoto [17], etc.…”

“…An important success was establishing the existence and uniqueness theorems by Lewy and others (see [10,3] for local aspects and [20] for global ones). The classical Monge integration method was modernized by Matsuda [15,16] and Morimoto [17], etc.…”