Fundamental solutions of real homogeneous cubic operators of principal type in three dimensions
To give an example of an explicit formula for a fundamental solution of an irreducible partial differential operator of higher order in more than two variables I. Fredholm represented the uniquely determined homogeneous and even fundamental solution of the elliptic operator \(\partial_1^4+\partial_2^4 +\partial_3^4\) by an elliptic integral containing the three variables \(x_1\), \(x_2\), \(x_3\) in a symmetric way (1908). Fredholm's starting point is his formula for a fundamental solution of a homogeneous, elliptic operator \(P(\partial_1, \partial_2,\partial_3)\) as an Abelian integral over the curve \(P(x_1,x_2,1)=0.\) His pupil N.~Zeilon aimed at doing the same for the real principal type operator \(\partial_1^3+\partial_2^3+\partial_3^3\); but, in fact, his representation is a sum of two elliptic integrals each of which contains the variables in an unsymmetric way. Using the addition theorem of Weierstraß's \(\wp\)-function the author derived a symmetric representation by an elliptic integral of the first kind [Math. Scand. 86, 273--287 (2000; Zbl 1043.35050)]. The paper under review extends this derivation and also the result to the operator \(P_a(\partial)= \partial_1^3+\partial_2^3+\partial_3^3+ a\partial_1\partial_2\partial_3\) (hyperbolic with respect to \((1,1,1)\) iff \(a<-1).\) Note that \(P_a(\partial)\) is Hesse's normal form of an irreducible operator in 3 variables and homogeneous of degree 3. Hence, an explicit symmetric representation of a fundamental solution of the class of homogeneous, third order differential operators in 3 variables by elliptic integrals of the first kind is found. The explicit result shows that the level sets of the fundamental solution are sixth degree algebraic surfaces. Also the 6th degree surface in which the singularities are located is derived. This was also done for operators in the Weierstraß normal form \(\partial_1^2 \partial_3-4\partial_2^3+g_2\partial_2\partial_3^2+g_3\partial_3^3\) in [\textit{J. Augé}, Collect. Math. 4, No. 1, 3--53 (1951; Zbl 0045.04704)]. The author's considerations are concrete applications and modifications of the \(``\)hyperbolic'' theory of \textit{M. F. Atiyah, R. Bott} and \textit{L. Gårding} [Acta Math. 124, 109--189 (1970; Zbl 0191.11203)]. His calculations are rather difficult -- using, e.g., Poincaré's residue map.
homogeneous third order partial differential operator
scientific article; zbMATH DE number 1541205
scientific article