The work of Hans Lewy (1904--1988) has had a profound influence in
the direction of applied mathematics and partial differential
equations, in particular, from the late 1920s. Two of the particulars
are well known. The Courant--Friedrichs--Lewy condition (1928), or CFL
condition, was devised to obtain existence and approximation results.
This condition, relating the time and spatial discretizations for
finite difference schemes, is now universally employed in the
simulation of solutions of equations describing propagation phenomena.
Lewy's example of a linear equation with no solution (1957), with its
attendant consequence that most equations have no solution, was not
merely an unexpected fact, but changed the viewpoint of the entire
field.
Lewy made pivotal contributions in many other areas, for example,
the regularity theory of elliptic equations and systems, the Monge--
AmpSre Equation, the Minkowski Problem, the asymptotic analysis of
boundary value problems, and several complex variables. He was among
the first to study variational inequalities. In much of his work, his
underlying philosophy was that simple tools of function theory could
help one understand the essential concepts embedded in an issue,
although at a cost in generality. This approach was extremely
successful.
In this two-volume work, most all of Lewy's papers are presented,
in chronological order. They are preceded by several short essays
about Lewy himself, prepared by Helen Lewy, Constance Reid, and David
Kinderlehrer, and commentaries on his work by Erhard Heinz, Peter Lax,
Jean Leray, Richard MacCamy, Fran?ois Treves, and Louis Nirenberg.
Additionally, there are Lewy's own remarks on the occasion of his
honorary degree from the University of Bonn.
to Volume 2.- [30] A note on harmonic functions and a hydrodynamical application.- [31] A theory of terminals and the reflection laws of partial differential equations.- [32] Asymptotic developments at the confluence of boundary conditions.- [33] Axially symmetric cavitational flow.- [34] On steady free surface flow in a gravity field.- [34A] An introduction to Riemann’s Work.- [35] Extension of Huyghen’s principle to the ultrahyperbolic equation.- [36] On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables.- [37] On the relations governing the boundary values of analytic functions of two complex variables.- [38] An example of a smooth linear partial differential equation without solution.- [39] On linear difference-differential equations with constant coefficients.- [40] Composition of solutions of linear partial differential equations in two independent variables.- [41] On the reflection laws of second order differential equations in two independent variables.- [42] On hulls of holomorphy.- [43] Atypical partial differential equations.- [44] Uniqueness of water waves on a sloping beach.- [47] On the definiteness of quadratic forms which obey conditions of symmetry.- [48] On the extension of harmonic functions in three variables.- [49] The wave equation as limit of hyperbolic equations of higher order.- [50] Sulla riflessione delle funzioni armoniche di 3 variablili.- [51] On a variational problem with inequalities on the boundary.- [52] On the nonvanishing of the jacobian of a homeomorphism by harmonic gradients..- [53] About the Hessian of a spherical harmonic.- [54] On the regularity of the solution of a variational inequality.- [55] On a minimum problem for superharmonic functions.- [56] On a refinement of Evans’ law in potential theory.- [57] On the partial regularity of certain superharmonics.- [58] On the smoothness of superharmonics which solve a minimum problem.- [60] On existence and smoothness of solutions of some non-coercive variational inequalities.- [61] On the coincidence set in variational inequalities.- [62] On the nature of the boundary separating two domains with different regimes.- [63] On analyticity in homogeneous first order partial differential equations.- [64] On the boundary behavior of holomorphic mappings.- [65] On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere.- [67] An inversion of the obstacle problem and its explicit solution.- [68] Expansion of solutions of t’Hooft’s equation. A study in the confluence of analytic boundary conditions.- [69] On conjugate solutions of certain partial differential equations.- [70] Über die Darstellung ebener Kurven mit Doppelpunkten.- [71] On free boundary problems in two dimensions.- [72] On the analyticity of minimal surfaces at movable boundaries of prescribed length.- [73] On atypical variational problems.