Variational principles for linear initial-value problems

Introduction. Most of the boundary-value problems of mathematical physics are characterized by variational principles which assert that a function u satisfies such a problem if and only if a given functional is stationary at u. However, there does not seem to exist in the treatise literature any such variational characterization of the usual initial-value problems of interest. As a matter of fact Washizu [1] conjectured that in the case of the transient heat conduction problem such a principle is not possible since the problem is not self-adjoint. In this note, variational principles of the foregoing type are derived for the initial-value problems associated with the wave and heat conduction equations, thus proving Washizu's conjecture incorrect. The method of deducing such variational principles consists of two steps. First we reduce the initial-value problem to an equivalent boundary-value problem by replacing the relevant partial differential equation with an equivalent integro-differential equation which contains the initial conditions implicitly. Second, we derive a variational principle for this reduced problem. The prime ingredient in the derivation of this variational principle is the use of convolutions. The techniques employed in this paper may be used to generate variational principles for a large class of initial-value problems (e.g. the initial-value problem associated with the time-dependent Schrodinger equation in quantum mechanics). Variational principles of this type for linear elastodynamics were given previously [2]. 1. Preliminary definitions. Notation. Henceforth R will denote a closed and bounded region in n-dimensional Euclidean space with the interior R and the boundary B, while x = (x-, , x2 , ■ ■ ■ , xn) is a generic point of R. Moreover V is the usual (redimensional) gradient operator and V2 is the Laplacian operator. Throughout this paper we deal with real-valued functions of position x and time t whose n + 1 dimensional domain of definition is the Cartesian product R X [0, °°) of the region R with the time interval [0, <»). We use the standard notation for the convolution of two such functions of position and time, i.e.