A class of second order boundary value problems concerning the capillarity theory (Q2711388)

The author studies the existence of solutions to the boundary problem NEWLINE\[NEWLINE-\biggl(r(t)\partial \varphi\bigl(u'(t) \bigr)\biggr)'+ q(t)Au(t)\ni f(t), \quad \text{a.e. on }(0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\biggl[r \partial\varphi (u')_{|t=0}, \quad -r(T) \partial\varphi \bigl(u'(T) \bigr)\biggr] \in\partial l\bigl(u(0), u(T)\bigr),NEWLINE\]NEWLINE in a Hilbert space \(H\) under the following main assumptions:NEWLINENEWLINENEWLINE(i) \(A:D(A) \subseteq H\to H\) is a maximal monotone operator;NEWLINENEWLINENEWLINE(ii) \(\varphi: H\to(-\infty, \infty]\), \(\ell:H \times H\to (-\infty,\infty]\) are lower semicontinuous, convex and proper functionals, with \(\partial\varphi\) and \(\partial \ell\) their subdifferentials;NEWLINENEWLINENEWLINE(iii) \(r\in C^1(0,T]\), \(q\in L^\infty (0,T]\), \(r(t)>0\), \(q(t)>0\) for \(t\in(0,T]\);NEWLINENEWLINENEWLINE(iv) \(f\in L^2(0,T;H)\). NEWLINENEWLINENEWLINEThis type of problems occurs in capillarity theory. Variational approach is the main tool in this investigation.