On the Relationship Between Fundamental Solutions of Elliptic and Parabolic Equations

Abstract

The fact that the integral of a fundamental solution of a three-dimensional heat equation from zero to innity is a fundamental solution of the Laplace equation was used for the rst time by Tikhonov [1] in the study of boundary value problems. In the same paper, Tikhonov proved a similar formula for the Green function of the Dirichlet problem. For parabolic systems of general form, Eidel’man [2, p. 145] proved conditions under which a fundamental solution matrix of an elliptic system can be obtained as a (regularized) integral with respect to time t from zero to innity of a fundamental matrix of the corresponding parabolic system. For systems with constant coecients, these conditions were expressed via the roots of equations composed of the coecients of the parabolic system. In the case of variable coecients, the formula was proved under the assumption that the fundamental solution matrix of the parabolic system admits an estimate ensuring the convergence of the integral of the fundamental solution matrix from zero to innity. Fundamental solution matrices of elliptic systems were constructed and estimated in the above-mentioned cases with the use of that formula. In the present paper, we obtain a new relation between fundamental solutions of elliptic equations with variable coecients inR n and fundamental solutions of the corresponding parabolic equations. In particular, this formula implies [see formula (21) below] that a fundamental solution of an elliptic equation in the entire R n (if it exists at all) coincides, neglecting a smooth additive term and up to the sign, with the integral of a fundamental solution of the parabolic equation from 0 to t for any t> 0. We think that our formula can be useful in various issues related to fundamental solutions of elliptic equations. In the present paper, we show that, in particular, this formula permits us to obtain new results concerning the principal fundamental solution in the sense of Giraud. Giraud (see [3, 4]) introduced the notion of a principal fundamental solution and proved its (