Extension of Fedorov's and Rado's theorems to solutions of the heat-conduction equation (Q5950876)

The well-known Fedorov-Radó theorem [see \textit{V. S. Federov}, Ivanovo-Voznesensk, Bull. Inst. polytechn. 1, 45-56, 139 (1919; JFM 48.1376.01); \textit{T. Radó}, Math. Z. 20, 1-6 (1924; JFM 50.0255.02)] is extended to the solutions of heat-conductivity equation. Function \(f(x)\) is twice continuously differentiable in variables \(x_k\) \((k=1,2,\dots,n-1)\), continuously differentiable in variable \(x_n\) in \(D\backslash E\), \( D\in \mathbb{R}^n\), and is a solution of the equation \( Lu=0 \) in \(D\), where \( L = \partial_n - \sum_{k=1}^{n-1}\partial_k^2 \) is the heat conductivity operator, and is called the \(L\)-function in \(D\). In particular, the following theorem is proved. If the set \( E\subset D\) is everywhere discontinuous, the function \(f(x)\) is continuous in \(D\), \( f(x) = 0 \) for any \( x\in E \) and for any closed hypersurface~\( S\subset D\backslash E\) \[ \begin{multlined}\sum_{k=1}^{n-1} (-1)^{k-1}\int_S \partial_kf(x) dx_1 \wedge \dots \wedge dx_{k-1} \wedge dx_{k+1} \wedge\dots\wedge dx_n\\ {}=(-1)^{n-1}\int_S f(x) dx_1 \wedge \dots \wedge dx_{n-1},\end{multlined} \] then \(f\) is the \(L\)-function in \(D\).