During the years when Lewy held a position at Göttingen he was awarded two Rockefeller Foundation Fellowships, the first allowing him to spend session 1929-30 at the University of Rome and second allowing him to spend session 1930-1931 at the University of Paris. These years were one in which he produced some outstanding mathematics. With Courant and Friedrichs he wrote Über die partiellen Differentialgleichungen der mathematischen Physik Ⓣ which appeared in Mathematische Annalen in 1928. In this paper criteria are given for determining conditions which guarantee the stability of numerical solutions of certain classes of differential equations. This work proved even more important after the advent of computers when the stability of numerical methods became crucial. In the following year he published, in the same journal, Neuer Beweis des analytischen Charakters der Lösungen elliptischer Differentialgleichungen Ⓣ.
M Protter, J L Kelley, T Kato, and D H Lehmer, in an obituary of Lewy, write about his achievements up to 1933:-
... he published a series of fundamental papers on partial differential equations and the calculus of variations. He solved completely the initial value problem for general non-linear hyperbolic equations in two independent variables. On the basis of this, and using the daring idea of converting an elliptic equation into a hyperbolic one by penetrating into the complex domain, he developed a new proof of the analyticity of solutions of analytic elliptic equations in two independent variables, one which far exceeded the known proof in its elegance and simplicity. He proved the well-posedness of the initial value problem for wave equations in what is now called Sobolev spaces two decades before these spaces became a common tool for specialists. The revolutionary character of these works is reflected in the fact that J Hadamard, a world authority at that time, devoted a special appendix to Lewy's theory in his newly published book on the Cauchy Problem (1932).On 30 January 1933 Hitler came to power and Lewy realised that he could not remain in Germany and that he had to emigrate. He went to the United States where he was appointed to Brown University for the two years 1933-35, and he was then appointed to the University of California at Berkeley as a lecturer in 1935. His career at Berkeley saw him promoted regularly with appointment as Assistant Professor in 1937, Associate Professor in 1941, and then full professor in 1945. During the Second World War he made his contribution to the war effort with an appointment at the Aberdeen Proving Ground. He married Helen Crosby on 9 June 1947 and they had one son, Michael R Lewy. Lewy retired in 1972 but continued to produce deep mathematics. He died after a brief illness.
Manfred Kracht, reviewing  and , writes that:-
Lewy's work is ... extremely original and inventive: He created many ideas which have opened new fields and are still waiting to be developed or extended in the future.Nirenberg  lists Lewy's mathematical papers under the following topics: (i) partial differential equations involving existence and regularity theory for elliptic and hyperbolic equations, geometric applications, approximation of solutions; (ii) existence and regularity of variational problems, free boundary problems, theory of minimal surfaces; (iii) partial differential equations connected with several complex variables; (iv) partial differential equations connected with water waves and fluid dynamics; (v) offbeat properties of solutions of partial differential equations. Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampère equations (two papers, the first in 1935, the second two years later), and On differential geometry in the large : Minkowski's problem (1938). Minkowski's problem is to construct a convex surface in three dimensional space that realises a given curvature as a function of the direction of the normal. He published Water waves on sloping beaches in 1946 studying the progressive wave on sloping beaches in two dimensions neglecting compressibility and assuming zero viscosity. The dock problem, written jointly with Friedrichs two years later, gives an explicit solution for the dock problem over a fluid of infinite depth. The solution is given by the sum of two integrals of Laplace type taken over a complex path of integration.
Another paper of major importance which Lewy published in 1951 was On minimal surfaces with partially free boundary which examines the continuation of minimal surfaces across analytic boundary arcs. His paper An example of a smooth linear partial differential equation without solution (1957) gave a simple partial differential equation which has no solution, a result which had a substantial impact on the area. This work led Lewy to two later papers on the theory of functions of several complex variables for which he received the Steele Prize of the American Mathematical Society in 1979.
Lewy received many honours for his mathematical contributions. He was elected a member of the National Academy of Sciences (United States), the American Academy of Arts and Sciences, and the Accademia dei Lincei (Rome). We mentioned above that the American Mathematical Society awarded him its Steele Prize in 1979. In 1986 he was awarded the prize of the Wolf Foundation along with Kodaira. In 1986 he received an honorary doctorate from the University of Bonn.
Lewy had wide-ranging talents and interests both within and outside mathematics. He loved music and languages, demonstrated an energy and enthusiasm for everything he undertook, showing a modesty, lack of pretension, and flair for philosophising. M Protter, J L Kelley, T Kato, and D H Lehmer write:-
He was noted in Berkeley for being a warm person with a fine sense of humour.They also illustrate his character with the following episode:-
[He] was known as a person of integrity and strong moral principles. In 1950, he refused to sign a special loyalty oath imposed on the faculty by the University of California's Board of Regents; for this reason he and a number of other professors were fired. They were later vindicated and reinstated when the courts determined that taking the oath would have violated their civil rights.
Article by: J J O'Connor and E F Robertson