A note on the strong maximum principle for elliptic differential inequalities

AbstractWe consider the strong maximum principle and the compact support principle for quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. This allows us to give necessary and sufficient conditions for the validity of both principles

The strong maximum principle revisited

AbstractIn this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g. (i)two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);(ii)the exterior Dirichlet boundary value problem (Section 5);(iii)the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);(iv)Euler–Lagrange inequalities on a Riemannian manifold (Section 9);(v)comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). The case of p-regular elliptic inequalities is briefly considered in Section 11

A new critical curve for a class of quasilinear elliptic systems

We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.Comment: 28 page

Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities

We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given

Nonlinear Hodge maps

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.Comment: 20 page

A Fresh Look at Entropy and the Second Law of Thermodynamics

This paper is a non-technical, informal presentation of our theory of the second law of thermodynamics as a law that is independent of statistical mechanics and that is derivable solely from certain simple assumptions about adiabatic processes for macroscopic systems. It is not necessary to assume a-priori concepts such as "heat", "hot and cold", "temperature". These are derivable from entropy, whose existence we derive from the basic assumptions. See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA

The Legacy of Theodore Leschetizky as Seen through His Pedagogical Repertoire and Teaching Style

Theodore Leschetizky's singular pianistic legacy survives to this day because of his revolutionary pedagogical methods and his compositions for the piano repertory. The amalgamation of these two aspects formed his distinctive contributions to the fields of piano and piano pedagogy and left an indelible mark on the history of the instrument. His students lead an impressive list of the greatest artists of the previous century, each influencing the evolution of pianism with their own remarkable style and personality. While Leschetizky was arguably without peer as a pedagogue, many pianists today are unaware of the vast number of compositions that he wrote. These pieces were intended not only for the concert stage, but also as a very specific pedagogical repertoire that he used within his own teaching studio. This repertoire comprises a vital component of the Leschetizky legacy, albeit one which is often slighted in comparison. It is imperative that the pianists of our current generation understand the dual aspects of his contribution to our art form, in order to fully grasp the way in which he has changed the face of pianism. The purpose of this dissertation and lecture recital is to enumerate the various aspects that constitute the dual components of Leschetizky's pianistic legacy. For pedagogues of the current generation, it is of vital importance that we understand not only our own personal pedagogical lineage, but the various other individuals that, through their contributions, have led us to where we are in our understanding of the instrument. What is needed in the current research on this subject is one individual source that not only documents the characteristics of a pedagogical genius, but explores the legacy he left for future generations through documented accounts of his students and the examination of his own unfamiliar, pedagogical repertoire for the piano