The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

This lecture presents some results by L. Caffarelli, Y. Y. Li, and the speaker, from [2], [3] and [4]. They concern extensions to singular solutions of various forms of the maximum principle; they consist of some small technical results.

We present several results, including some remarks on the Hopf Lemma.

In [1], Theorem 3, the authors proved, in one dimension, a generalization of the Hopf Lemma, and the question arose if it could be extended to higher dimensions. In this paper we present two conjectures as possible extensions, and give a very partial answer. We write this paper to call attention to the problem.

We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes.

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,\alpha}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

This, and its sequel, concern some variations of a classical theorem of A.D. Alexandrov and teh Hopf Lemma.

. We prove regularity up to the boundary of the solution of an unusual variational problem arising in mathematical finance.

We study the distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations.

Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler metric). We prove that the distance function from $\partial \Omega$ is in $C^{k,\alpha}_{loc}(G\cup \partial \Omega)$, $k\ge 2$ and $0<\alpha\le 1$, if $\partial \Omega$ is in $C^{k,\alpha}$.

was one of the principal contributors to the field of partial differential equations (PDE) for the past half century.Her list of publications is remarkable, numbering over 300, and they cover all subjects of PDE, with the exception of microlocal analysis.She made deep contributions to linear and nonlinear theory as well as in applications, especially in elasticity theory.I will mention just a few of the topics on which she worked.Other people will surely mention more.She did fundamental work on nonlinear hyperbolic equations: conservation laws.Many papers on elliptic and parabolic equations, including mixed boundary conditions; the averaging method; homogenization, also for rapidly oscillating boundaries; domains with cavities; spectral problems; biharmonic equations, and boundary estimates.She obtained important estimates in nonsmooth and unbounded domains.The work with which I am most familiar, since it overlaps with some of my own, is work on asymptotic behavior of solutions of nonlinear elliptic and parabolic equations in a cylinder and work on degenerate second-order elliptic equations.For the latter, her excellent 1987 book with E. V. Radkevich is a basic reference, as is her 1997 book on boundary-layer theory (with co-authors).Oleinik's work in elasticity theory is beautiful; it includes extensions of Korn's inequality in various situations, and the Saint-Venant principle.In all of her work, inequalities play a major role.On a more personal note, Olga and I first met, as I recall, at the International Congress in Edinburgh in 1958.Subsequently we met frequently, in Russia, France, Italy, and the U.S., and we discussed mathematics a great deal.For many years she was one of the few mathematicians in the Soviet Union who could travel rather easily.I greatly admired her originality and strong analytic technique.She was a forceful person, and we all miss her strong personality as well as her mathematics.

Communications on Pure and Applied MathematicsVolume 56, Issue 7 p. 892-925 Estimates for elliptic systems from composite material† YanYan Li, YanYan Li yyli@math.rutgers.edu Rutgers University, Department of Mathematics, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019Search for more papers by this authorLouis Nirenberg, Louis Nirenberg nirenl@cims.nyu.edu Courant Institute, 251 Mercer Street, New York, NY 10012-1185Search for more papers by this author YanYan Li, YanYan Li yyli@math.rutgers.edu Rutgers University, Department of Mathematics, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019Search for more papers by this authorLouis Nirenberg, Louis Nirenberg nirenl@cims.nyu.edu Courant Institute, 251 Mercer Street, New York, NY 10012-1185Search for more papers by this author First published: 23 April 2003 https://doi.org/10.1002/cpa.10079Citations: 149 † Dedicated to the memory of Jürgen Moser AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume56, Issue7Special Issue: First of Two Special Issues Dedicated to the Memory of Jürgen K. MoserJuly 2003Pages 892-925 RelatedInformation

We give necessary and sufficient conditions for a smooth, generic, differential oneform ω on R n to decompose into a sum ω = a 1 du 1 + . . .+ a k du k , where the functions a are positive and the u convex (or quasi-convex) near the origin.