engleski

On some new properties of solutions of nonlinear PDE-s

We describe the problem of local {; ; \em control of essential supremum}; ; of weak solutions for a class of $p$-Laplace equations on an open domain, introduced by Mervan Pa\v si\'c (University of Zagreb), see \cite{; ; kpz}; ; . Some of the consequences of the main result are explicit lower bounds on Agmon-Douglis- Nirenberg constants in terms of the inner radius of domain, variational estimates of oscillation of weak solutions, and generating infinite oscillation of solutions in a given point. Next we describe the problem of measuring the set of singular points of weak solution of the simplest $p$-Laplace equation by means of fractal dimensions. To this end the notion of {; ; \em singular dimension of a space}; ; (or set) of Lebesgue measurable real functions is introduced as the supremum of Hausdorff dimension of singular sets of functions in the space, see \cite{; ; sing}; ; , \cite{; ; singh}; ; , \cite{; ; besov}; ; , \cite{; ; gensing}; ; . If the supremum is achieved, the function is said to be {; ; \em maximally singular}; ; in the space. Some clasess of linear and nonlinear Laplace equations are shown to possess maximally singular weak solutions. We also consider the Hopf bifurcation of a class of completely integrable nonlinear Schr\"odinger Cauchy problems on a bounded domain and study the complexity of solutions at Hopf bifurcation by means of box dimension of trajectories and Minkowski content of their coordinate projections with respect to eigenfunctions of the Laplace operator. We compute the {; ; \em sequence of Minkowski contents}; ; (or {; ; \em Minkowski sequence}; ; assigned to a trajectory) in terms of Fourier coefficients of the initial function and of the eigenvalues of the Laplace operator. We also study the asymptotic properties of the Minkowski sequence of the trajectory. The box dimension of $n$-chirps and spiral chirps related to the trajectory are computed. This is a continuation of work on spiral trajectories of planar vector fields undertaken jointly with Vesna \v Zupanovi\'c, see \cite{; ; zuzu}; ; and \cite{; ; aarhus}; ; , now in infinite-dimensional setting. \begin{; ; thebibliography}; ; {; ; 10}; ; \let\small=\rm \let\normalsize=\rm \bibitem{; ; kpz}; ; L.\ Korkut, M.\ Pa\v si\'c, D.\ \v Zubrini\'c, Some qualitative properties of solutions of quasilinear elliptic equations and applications, {; ; J.\ Differential Equations}; ; , {; ; 170}; ; (2001), no. 2, 247--280. \bibitem{; ; sing}; ; D.\ \v Zubrini\'c, Singular sets of Sobolev functions, {; ; C.\ R.\ Acad.\ Sci., Analyse math\'ematique}; ; , Paris, S\'erie I, {; ; 334}; ; (2002), 539--544. \bibitem{; ; singh}; ; L.\ Horvat, D.\ \v Zubrini\'c, Maximally singular Sobolev functions, J.\ Math.\ Anal.\ Appl.\ {; ; 304}; ; (2005), no. 2, 531--541. \bibitem{; ; besov}; ; D.\ \v Zubrini\'c, Maximally singular functions in Besov spaces, Arch. Math.\ {; ; 87}; ; (2006), 154-162. \bibitem{; ; gensing}; ; D.\ \v Zubrini\'c, Generating singularities of solutions of $p$- Laplace equations on fractal sets, Rocky Mountain J.\ Math., 39 (2009) , 1 ; 359-366 \bibitem{; ; zuzu}; ; D.\ \v Zubrini\'c, V.\ \v Zupanovi\'c, Fractal analysis of spiral trajectories of some planar vector fields, Bulletin des Sciences Math\'ematiques, 129/6 (2005), 457-485. \bibitem{; ; fdd}; ; V.\ \v Zupanovi\'c, D.\ \v Zubrini\'c, Fractal dimensions in dynamics, in: Encyclopedia of Mathematical Physics, eds.\ J.-P.\ Fran\ced{; ; c}; ; oise, G.L.\ Naber and Tsou S.T.\ Oxford: Elsevier, (2006), vol 2, 394--402. \bibitem{; ; aarhus}; ; V.\ \v Zupanovi\'c, D.\ \v Zubrini\'c: Recent results on fractal analysis of trajectories of some dynamical systems, {; ; \em FUNCTIONAL ANALYSIS IX}; ; - Proceedings of the Postgraduate School and Conference held at the Inter-University Centre, Dubrovnik, Croatia, 15-23 June, 2005.\ pp.\ 148.\ / G.\ Muic, J.\ Hoffmann-Jorgensen (eds.). Aarhus, Denmark: University of Aarhus, Department of Mathematical Sciences, 2007. 126-140.

p-Laplace equation; weak solution; control of solutions; singular set; fractal set

nije evidentirano