A noncharacteristic cauchy problem for linear parabolic equations II: a variational method

Many inverse heat conduction problems lead us to consider the following noncharacteristic Cauchy problem for parabolic equations of the form “surface temperature” “surface heat flux” where p is an elliptic operator, ϕ and g are given functions. This problem is well-known to be severely ill-posed, and up to now there have been many approaches for solving it in a stable way. However, most of them need a supplementary condition: either the initial condition, or a boundary condition, etc⃜In this paper a variational method for this problem is suggested. In contrast to the other works, in the paper the initial condition is not assumed to be known. A short discussion on using the gradient methods is also given.