Degenerate second-order differential equations in Hilbert spaces
type: Item
instance of: scholarly article
title: Degenerate second-order differential equations in Hilbert spaces
zbMATH Open document ID: 0865.34046
DOI: 10.1007/BF02104822
author: V. I. Nalimov
published in: Siberian Mathematical Journal
publication date: 1997-07-07T00:00:00Z
review text: This article deals with the weak solvability properties of an ordinary differential equation \[ (A_2(t)u'(t))'+ A_1(t)u'(t)+A_0(t)u=f(t)\quad (0<t<T) \] under one of the following additional boundary value conditions: \[ u(0)=0\quad (A_2(0)\leq 0,\;A_1(T)\geq0), \] \[ u(0)=0,\quad A_2(T)u'(T)=0\quad (A_2(0)\leq0,\;A_1(T)\leq 0), \] \[ u(0)=0,\quad A_2(0)u'(0)=0\quad (A_2(0)\geq0,\;A_1(T)\geq 0), \] and \[ u(0)=0,\quad A_2(0)u(0)=0,\quad A_2(T)u'(T)=0\quad (A_2(0)\geq 0,\;A_2(T)\leq 0). \] The operators \(A_0(t)\), \(A_1(t)\), \(A_2(t)\) act between separable Hilbert spaces \(V_k\) \((k=0,1,2)\) and their adjoints \(V_k'\) with respect to a scalar product \((\cdot,\cdot)\), \(V_0\subset V_1\subset V_2\) and \(V_0\) is dense in \(V_2\). The operator functions \(A_0(t)\) and \(A_2(t)\) are strictly differentiable, \(A_1(t)\) is strictly continuous; the operators \(A_2(t)\) are selfadjoint; the operators \(A_0(t)\) are positive definite; and the functional \[ J(\phi)=\int^T_0((A_1+\textstyle{{1\over 2}} A_2')\phi',\phi')+\textstyle{{1\over 2}} ((A_0-A^*_0)\phi',\phi)-\textstyle{{1\over 2}} (A_0'\phi,\phi)dt \] satisfies the inequality \(J(\phi)\geq \gamma|\phi'|^2_{L_2(0,T;V_1)}\), \(\gamma>0\). The basic results are theorems about the existence of weak solutions and some ``energetic'' inequalities both in the interior of \((0,T)\) and in the whole \((0,T)\); the proofs are based on classical regularizations by means of the Lax-Milgram selfadjoint operator \(A\) which corresponds to the problem under consideration. In the end of the article some simple examples for ordinary and partial differential operators are considered.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 35R20
zbMATH DE Number: 917074
zbMATH Keywords: energetic inequalities
zbMATH Keywords: weak solutions
zbMATH Keywords: Lax-Milgram selfadjoint operator
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: Q4177987
description: scientific article; zbMATH DE number 917074