Degenerate second-order differential equations in Hilbert spaces
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.
scientific article; zbMATH DE number 917074