Non Linear Elliptic Theory and the Monge-Ampere Equation
نویسنده
چکیده
The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ainpere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations. 2000 Mathematics Subject Classification: 35J15, 35J20, 35J70. When learning complex analysis, it was a remarkable fact tha t the real part u of an analytic function, just because it satisfies the equation: « I I +Uyy = A U = 0 (Laplace's equation) is real analytic, and furthermore, the oscillation of u in anygiven domain U, controls all the derivatives of u, of any order, in any subset Ü, compactly contained in U. One can give three, essentially different explanations of this phenomena. a) Integral r epresenta t ions (Cauchy integral, for instance). This gives rise to many of the modern aspects of real and harmonic analysis: fundamental solutions, singular integrals, pseudo-differential operators, etc. For our discussion, an importan t consequence of this theory are the Schauder and Calderon-Zygmund estimates. Heuristically, they say tha t if we have a solution of an equation
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