Investigation of the behavior of weak waves in the solution of a vector variational wave equation (Q1048381)

Problems stated in variational form arise in various physical theories, in particular, in field theory, hydrodynamics (wave propagation), and circuit analysis. In some cases, a smooth solution of such problems in a domain may have a weak discontinuity (a discontinuity of the derivative). In the present paper, the authors mainly deal with the case in which the onset of the discontinuity is caused by nonsmoothness of the boundary conditions. The following variational problem for the unknown function \(u(x): G\to\mathbb{R}^N\), \(x\) belongs to \(G\subset\mathbb{R}^n\), \[ J= \int_G F(x,u(x), p(x))\,dx\to \text{extr}\biggl(p={\partial u\over\partial x}\biggr), \quad B[u(x)]_{x\in \partial G}= 0\tag{1} \] is considered. The specific form of the boundary conditions \(B[u(x)]= 0\) depends on the type of the problem; one can specify the values of the function itself and its partial derivatives on various parts of the boundary. The functional (1) is considered on the set \(U= \{u^*(x), G_*\}\) whose elements are pairs \((u^*(x), G_*)\), where \(u^*(x)\) is a continuous function defined and twice piecewise continuously differentiable in a domain \(G_*\). In general, each function \(u^*(x)\) has its own domain \(G_*\). The Lagrangian \(F\) is assumed to be continuously differentiable sufficiently many times. Here, a quadratic Lagrangian and the corresponding variational wave equation are considered. The problem is to find necessary conditions for smooth branches of the solution on the boundary of the domains and to describe an algorithm for constructing singular surfaces.