Asymptotics of the energy integrals for problems with small parameter at higher derivatives (Q1902711)

Let \(\Omega\) be a subdomain of \(\mathbb{R}^n\) with smooth (of the class \(C^\infty\)) boundary and with compact closure \(\overline\Omega\). We consider the functional \[ J_\varepsilon(u)= \varepsilon^2(\Delta u, \Delta u)+ (\nabla u, \nabla u)- 2(f, u),\tag{1} \] where \(\varepsilon\) is a small positive parameter, \(\nabla=\text{grad}\), \(\Delta= \nabla\cdot \nabla\), \((\cdot, \cdot)\) is the scalar product in \(L_2(\Omega)\), \(u\) is an element of the Sobolev space \(W^2_2(\Omega)\), \(f\in L_2(\Omega)\). The function \(x\mapsto u(\varepsilon, x)\) realizing the minimum of the functional (1) on the set \(W^{2, 0}_2(\Omega)\) satisfies the boundary value problem \[ \begin{aligned} \varepsilon^2 \Delta^2 u(\varepsilon, x)- \Delta u(\varepsilon, x)= f(x), &\quad x\in \Omega;\tag{2}\\ u(\varepsilon, x)= (\partial u/\partial\nu)(\varepsilon, x)= 0, &\quad x\in \partial\Omega.\tag{3}\end{aligned} \] Its limit case \((\varepsilon= 0)\) is the Dirichlet problem for the Laplace operator \(- \Delta v(x)= f(x)\), \(x\in \Omega\); \(v(x)= 0\), \(x\in \partial\Omega\), and \(J_0(v)\) is the minimal value of the limit functional on \(W^{1, 0}_2(\Omega)\). Clearly, for any element \(w\) of the space \(W^{2, 0}_2(\Omega)\) there is valid the inequality \[ J_\varepsilon(w)\leq J_0(w)+ c_w \varepsilon^2,\quad c_w\geq 0.\tag{4} \] The article is dealing with an evaluation of the main terms of asymptotics of energy functionals (1). The authors apply to this end the Vishik-Lyusternik technique, i.e., the asymptotical series for the mentioned functional contains only integer powers of the parameter \(\varepsilon\). One of the consequences of the obtained asymptotical formula is the relation \(J_\varepsilon(u)- J_0(v)\sim c_f \varepsilon\) which shows that bound (4) does not withstand the passage to the minimum. The authors consider other boundary value problems for the operator \(\varepsilon^2 \Delta^2- \Delta\) and obtain analogous asymptotical formulas for a number of boundary value problems of theory of elasticity with intrinsic small parameters.