Cauchy problems on a characteristic conoid for the (conformal) Einstein vacuum equations and the Yang-Mills-Higgs equations
Let \(x= (x^0, x^1, x^2, x^3)\in R^4\), \(s^2= (x^1)^2+ (x^2)^2+ (x^3)^2\), \(s(x^1, x^2, x^3)\geq 0\), \(T> 0\); \(C= \{x; x^0= s\}\), \(Y= \{x; s\leq x^0\}\), \(C_T= \{x;s= x^0\leq T\}\), \(Y_T= \{x; s\leq x^0\leq T\}\). The author has studied the Cauchy problem on the characteristic: \(A^{\lambda\mu}(x,u) D_{\lambda\mu}u+ f(x,u,Du)= 0\), \(u= (u^I; I=1\sim N)\), \(\lambda,\mu= 0,1,2,3\), in \(Y\) with \(u= \phi(x)\) on \(C_T\). Conditions: \(A^{00}> 0\), \(\sum_{i,j=1\sim 3}A^{ij} X_iX_j< 0\), there exists \((a^I)\) such that \(A^{\lambda\mu}(0 (a^I))= \eta^{\lambda\mu}=\) (M. Minkowski); \(\phi(0)= (a^I)\), \(A^{00}(x; \phi)- 2\sum_{k=1\sim 3} A^{0k}(x; \phi)x^k/s+ \sum A^{ij}(x; \phi)x^i x^j/s^2= 0\) on \(C_T\); \(f(x,u,Du)\in C^{2m-3}\), \(m\geq 3\). He has proved the existence of the unique solution \(u=\overline u+ u_1\) in \(Y_{T_0}\) for a \(T_0\in (0,T]\), and has derived \(T= T_0\) under some conditions. He applies his results to the Einstein vacuum equations \(R_{\alpha\beta}(g)= 0\) from the viewpoint of the gravitational radiation in cosmology. If the supplemental gauge conditions given on \(C_T\) hold, the solution \(g\) of the Cauchy problem: \(-g^{\lambda\mu}\partial^2_{\lambda\mu} g_{\alpha\beta}/2+ f_{\alpha\beta}(x, g_{\mu\nu}, \partial_\delta g_{\mu\nu})= 0\) in \(Y\) with \(g_{\alpha\beta}|_C= h_{\alpha\beta}\) is the solution of the Einstein equations. He also tries to apply his results to the Yang-Mills-Higgs equation, etc.
scientific article; zbMATH DE number 1964635