Difficult relationships between functional analysis and mathematical physics
In order to point out the basic role played by the choice of the norm for stating properties in problems of analysis and mathematical physics, the author develops and discusses three examples. Firstly the Volterra integral equation: \(U(x,t)=\exp (-x^ 2)+\int^{t}_{1}(2/\tau -6\tau^ 5x^ 2)U(x,\tau)d\tau,\) having as solution the function: \(u(t)\equiv U(x,t)=t^ 2\exp (-t^ 6x^ 2)\) defined for \(t\in [1,+\infty)\) with values in a Banach space S. The author comes to the conclusion that assuming as space S the space with the norm \(\| \nu \| =\int^{1}_{-1}| \nu (x)| dx\) the above solution u(t) is asymptotically stable. At the contrary the statement is not true if the space S has the norm \(\| \nu \| =\max_{[-1,1]}| \nu (x)|.\) A second example refers to the ``well-posedness'' of the Cauchy problem for the Laplace operator. The last example refers to the ``principle of fading memory'' in visco- elasticity, with consideration of the integral equation in \(\epsilon\) (t): \(\sigma (t)=G(0)\epsilon (t)+\int^{t}_{t_ 0}G(t-\tau)\epsilon (\tau)d\tau.\) The assumed functional space \(S_{\gamma}\), for \(\epsilon\) (t) is: \(\epsilon (t)\in L^ 2_{loc},\quad \int^{+\infty}_{0}| \epsilon (-s)|^ 2\exp (-2\gamma s)ds<+\infty,\) and the particular value considered for G(s) is \(1/2+1/2 \exp (-s).\) The author comes to the conclusion that for \(\gamma\leq 1/2\) the uniqueness theorem holds for the said integral equation (with \(t_ 0=-\infty)\). At the contrary for \(\gamma >1/2\) the theorem is not true.
scientific article; zbMATH DE number 4165685