An impedance effect of a thin adhesive layer in some boundary value and transmission problems governed by elliptic differential equations (Q2806461)

The authors consider a class of boundary value and transmission second-order operational problems NEWLINE\[NEWLINE(u^{\delta})''(x)+Au^\delta(x)=g^{\delta}(x),\quad x\in]-1,0[\cup]0,\delta[\cup]\delta, 1+\delta [,NEWLINE\]NEWLINE NEWLINE\[NEWLINE u^{\delta}(-1)=f_-(u^{\delta})'(1+\delta)=f_+\quad u^{\delta}(0^-)=u^{\delta}(0^+)+\alpha^{\delta},\quad u^{\delta}(\delta^-)=u^{\delta}(\delta^+)+\beta^{\delta}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE p_-(u^{\delta})'(0^-)=p_0(u^{\delta})'(0^+)+a^{\delta},\quad p_0(u^{\delta})'(\delta^-)=p_+(u^{\delta})'(\delta^+)+b^{\delta} NEWLINE\]NEWLINE in some complex Banach space \(E\), \(A\) is a closed linear operator of domain \(D(A)\subset E\) which verifies the Krein's ellipticity condition, \(f_-\), \(f_+\), \(\alpha^{\delta}\), \(\beta^{\delta}\), \(a^{\delta}\), \(b^{\delta}\) are given in \(E\) and \(g\) is a function.NEWLINENEWLINEFor \(\delta\to 0\), the authors obtain a boundary value transmission problem set on a fixed domain. New results for the study of this problem in the framework of Hölder spaces are derived -- explicit representation of the solution and necessary and sufficient conditions at the interface for its optimal regularity using the semigroups theory and the real interpolation spaces.