"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.Comment: 23 pages, 3 figure

Reaction-diffusion problems on time-dependent Riemannian manifolds: stability of periodic solutions

We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions. Such problems are posed on compact submanifolds evolving periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is motivated by biological models on the effect of growth and curvature on patterns formation. The Ricci curvature plays an important role

Asymptotic behaviour of large solutions of quasilinear elliptic problems

The paper deals with the large solutions of the problems $\triangle u=u^p$ and $\triangle u= e^u.$ They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P-function to establish the asymptotic behaviour of the gradient

A priori bounds for a class of nonlinear elliptic equations and applications to physical problems

Upper and lower bounds for the solutions of a nonlinear Dirichlet problem are given and isoperimetric inequalities for the maximal pressure of an ideal charged gas are constructed. The method used here is based on a geometrical result for two-dimensional abstract surface

"Legal Questions of Art Auctions” (Rechtsfragen der Kunstauktion): Seminar held by the Europe Institute, University of Zurich and the Center of Art and Law, Zurich, 13 April 2011: 13 April 2011

Directed by Professor Dr. Kurt Siehr, Professor Dr. Wolfgang Ernst, and Dr. Andrea F. G. Raschèr, the seminar exposed the legal fundaments of art auctions and provided an overview of some underlying problems currently faced by practitioners and legal scholars. The seminar was followed by a panel discussion called the "Boon and Bane of Auction Houses for the Art Market,” gathering directors of auction houses as well as art market and art law expert

Symmetry of large solutions of nonlinear elliptic equations in a ball

Let $g$ be a locally Lipschitz continuous real valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of $-\Delta u+g(u)=0$ in a ball is radially symmetric

Blowup in diffusion equations: A survey

AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems