Franco Obersnel

Classical and Non-Classical Sign-Changing Solutions of a One-Dimensional Autonomous Prescribed Curvature Equation

Abstract We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problem , u(0) = 0, u(1) = 0, depending on the behaviour at the origin and at infinity of the function f. We consider solutions that are possibly discontinuous at the points where they attain the value zero.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

Abstract We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation . Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions

Abstract An elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.

Period two implies chaos for a class of ODEs

We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.

Some notes on weakly Whyburn spaces

Abstract We construct in ZFC two compact weakly Whyburn spaces that are not hereditarily weakly Whyburn, one of them is also sequential. We also construct a Hausdorff countably compact space and a Tychonoff topological group both of weight ω 1 that are not weakly Whyburn. We finally show that Whyburn and weakly Whyburn properties are not preserved by pseudo-open maps.

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Abstract We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space { - div ⁡ ( ∇ ⁡ u 1 - | ∇ ⁡ u | 2 ) = f ⁢ ( x , u , ∇ ⁡ u ) in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω .

Chapter 3 A Qualitative Analysis, via Lower and Upper Solutions, of First Order Periodic Evolutionary Equations with Lack of Uniqueness

\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.

Continuous images of H∗ and its subcontinua

The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

Classical and non-classical solutions of a prescribed curvature equation

Publisher Summary The method of lower and upper solutions is an elementary but powerful tool in the existence theory of initial and periodic problems for semilinear differential equations for which a maximum principle holds, even in cases where no special structure is assumed on the nonlinearity. This chapter shows that this method is quite effective for investigating the qualitative properties of solutions, at the same extent of generality for which the existence theory is developed.

Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

We give new sufficient conditions for a continuum to be a remainder of H . We also show that any non-degenerate subcontinuum of H∗ maps onto any continuum of weight ≤ω1 , thus generalizing a result of D.P. Bellamy.