Hadamard's fundamental solution and conical refraction (Q2759791)

The author's aim is to prove that ``Hadamard's fundamental solution for the partial differential operator NEWLINE\[NEWLINE\biggl(\partial_t^2-\frac 1{\sigma_1}\Delta\biggr) \biggl(\partial_t^2-\frac 1{\sigma_3}(\partial_1^2+\partial_2^2)- \dfrac 1{\sigma_1}\partial_3^2\biggr),\quad \sigma_1>\sigma_3>0, \tag{U}NEWLINE\]NEWLINE is singular at both sheets of its characteristic conoid defined by the equations \(t^2-\sigma_1(x_1^2+x_2^2+x_3^2)=0,\;t^2-\sigma_3(x_1^2+x_2^2)- \sigma_3x_3^2.\)'' \((U)\) is the determinant operator - up to a factor \(\partial_t^2\) - of Maxwell's system for the electromagnetic vector \(u\) in uniaxial crystals characterized by two different dielectric constants (multiples of \(\sigma_1\) and \(\sigma_3).\) Physically \(u\) describes double refraction (and not conical refraction). The author is not aware of the explicitly known fundamental solution of \((U)\) which clearly exhibits the singular support of the fundamental solution with support in \(t\geq 0\) [cf. p. 30, 8. Beispiel in \textit{P. Wagner}, Parameterintegration zur Berechnung von Fundamentallösungen. Diss. Math., Warszawa 230 (1984; Zbl 0534.35008)].NEWLINENEWLINENEWLINEThe paper reports on some general facts on the construction of Hadamard's fundamental solution. Most concepts are not defined (e.g., \(F(t,x)\) in equation (14), p.~264; \(C(x)\) in Theorem 3, p.~266; and a Dirac-type distribution on p.~269), six theorems are stated without proofs.NEWLINENEWLINENEWLINEAn explicit expression of the fundamental matrix of electrodynamics in uniaxial crystals is given in \textit{N.~Ortner} and \textit{P.~Wagner} [Remarks on fundamental matrices (Green's tensors) in elastodynamics, piezoelectricity and crystal optics. In: Functional-analytic and complex methods, their interactions, and applications to partial differential equations. H. Florian (ed.) et al., World Scientific, New Jersey, 2001, 178-187 (2001); Proposition 2 on p. 185, 186].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].