Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type (original) (raw)

1985, Rendiconti Del Circolo Matematico Di Palermo

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)⊂H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0tu(0)=u 0∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu 0 whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx 1,x 2,…,x N tou(T 1),…,u(T N) with 0T, T N and positive weightsP i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0t)E. The function β(t) is larger thant/m whentk andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} ) , which β(t)=t/m on [T k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q ∞(u 0)=max{‖u(T i)−x i‖, 1≤i≤N}.