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Extending Peano derivatives: necessary and sufficient conditions

Hans Volkmer — 1999

Fundamenta Mathematicae

The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Albert J. MilaniHans Volkmer — 2011

Applications of Mathematics

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u t t + 2 u t - a i j ( u t , u ) i j u = f corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation - a i j ( 0 , v ) i j v = h . We then give conditions for the convergence, as t , of the solution of the evolution equation to its stationary state.

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