Minimal solutions to a class of parabolic partial differential equations
T. Guinn, Edward M. Landesman · Proceedings of the American Mathematical Society · 1969
2. Statement of the problem. Let T be a closed interval [to, t'] of the real line. Suppose Q is an open set in n-dimensional Euclidean space En and let i2 be a compact subset of R. Let 9j be the set of real-valued functions u(t, x) defined on TXQ and absolutely continuous on T for almost all x such that for some fixed integer k, the function u and all partial derivatives of u with respect to x of order less than or equal to k are in 22(Q) for fixed t and in ?2(T) for fixed x. For each t(ET, define the k-Dirichlet norm2 of u to be