Contact problems for linear equations of mathematical physics (Q2713920)

The author presents a review of recent results on the so-called contact problems for partial differential equations of the form NEWLINE\[NEWLINE \vec a \nabla u = u_{yy},\quad \vec a = \vec a(x,y), NEWLINE\]NEWLINE which arises in many physical processes, such as propagation of heat, filtration, reciprocal flows in a boundary layer, and others. The unknown \(u\) may accordingly stand for the concentration of a passive scalar in a liquid, density, or temperature. To be specific, for the study of contact problems, the author is concerned throughout the article with the following statement: On \(D = \Omega\times (0,T)\), \(\Omega\subset\mathbb R\) (or \(\Omega\subset\mathbb R^n\)) find a solution to the equation NEWLINE\[NEWLINE fu_t = Lu, NEWLINE\]NEWLINE where \(L\) is a strong elliptic equation of the second order. The class of functions \(f(x,t)\) includes functions which change their sign on \(\Omega\). In particular, the following three types of \(f\) are considered: a) \(f = f(t)\in C[0,T]\), where \(f(t_0) = 0\), and \(f(t) > 0 (< 0)\) for \(0 < t < t_0\) \((t_0 < t < T)\); b) \(f = f(x)\) is a positive function discontinuous at \(x = 0\), \(0\in\Omega\); c) \(f = \operatorname{sgn}x\). For these types of functions, the author states and studies various nonclassical boundary value problems. Existence and uniqueness theorems are presented for various classes of function spaces.