We consider the problem of finding a couple of solutions satisfying the following conditions (4) and (5) for a couple of two uniformly elliptic partial differential operators and of order in a non regular situation.
Si considera il problema al contorno in , , dove è un aperto limitato e connesso ed è un parametro reale. Si prova che, se è «superlineare» ed è abbastanza piccolo, il problema precedente ha almeno tre soluzioni distinte.
In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks...). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro.Finally, we reformulate some uncertainty principles in terms of properties of the free heat and shrödinger equations.
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data , where and , , or , or . Moreover, if , or if , or if and we show that the Cauchy problem is unconditionally wellposed in . Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal...