Contrary to widespread perception, there has ever since 1994 been a unified, general, that is, type independent theory for the existence and regularity of solutions for very large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, see [21,22], and for further developments [1-3,47-56,58,64-66]. This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs. All the solutions obtained have a blanket, universal, minimal regularity property, namely, they can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains. It is important to note that the use of the order completion method does not require any monotonicity conditions on the nonlinear systems of PDEs involved. One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy "linear versus nonlinear" PDEs, treating both cases equally. Furthermore, the order completion method does not introduce the dichotomy "monotonous versus non-monotonous" PDEs. None of the known functional analytic methods can exhibit such a powerful general performance, since in addition to topology, such methods are significantly based on algebra, and vector spaces do inevitably differentiate between linear and nonlinear entities. The power of the order completion method is also shown in its ability to solve equations far more general than PDEs, and give in fact necessary and sufficient conditions for the existence of their solutions, as well as explicit expressions for the solutions obtained. In the case of PDEs, another advantage of the order completion method is that in treating initial and/or boundary value problems it avoids the considerable additional difficulties which the usual functional analytic methods encounter. Nevertheless, there are certain basic connections and similarities between the usual functional analytic methods in solving PDEs, and on the other hand, the order completion method. And in fact, the ancient equation x^2 = 2 which had two and a half millennia ago created such a terrible conundrum for Pythagoras, has ultimately been solved by a simple version of the order completion method. Indeed, its irrational solution was obtained in the set R of real numbers, while that set itself was obtained by the Dedekind order completion of the set Q of rational numbers, when that latter set is considered with its natural order relation.
About the Author: Emer. Prof. Elemer Elad Rosinger has been visiting, or spent time for longer periods, at numerous universities on five continents. For a while by now, he has been a member of the Department of Mathematics and Applied Mathematics at the University of Pretoria, South Africa. Originally, he is from Transylvania, in Europe. Prof. Rosinger's research is in a number of fields in mathematics and physics. Among others, he introduced two nonlinear theories for solving large classes of nonlinear PDEs. One of them is listed under 46F30 by the American Mathematical Society. The second one is based on the order completion of spaces of smooth functions. He also gave in 1998 the first complete solution to Hilbert's Fifth Problem, which was outstanding ever since its formulation in 1900. Prof. Rosinger has published about two hundred papers and a dozen books, most of them in science. Some of the papers and books may usually be seen as dealing with philosophy, if only one could ever give a good enough definition for that human venture ... But then, the reach and ability of definitions are very limited, indeed. As for instance illustrated by the saying: "I can't tell what a good wine is, but certainly recognize one when I drink one ..." Fortunately, mathematics is one of those few realms of both great charm and importance where many things, nearly all things, can be defined precisely ... And still, it is quite possible to get a genuine and better kind of "high", even if not exactly drunk, by mathematics ... eerosinger@hotmail.com