(untitled) (Analysis in Theory and Applications, 2020)
We explore minimization problems of the form Inf { ^1_0 |u'|+ ^k_i=1 |u(a_i) - f_i|^2 + α ^1_0 |u|^2}, where u is a function defined on (0,1), (a_i) are k given points in (0,1), with k≥ 2, (f_i) are k given real numbers, and α ≥0 is a parameter taken to be 0 or 1 for simplicity. The natural functional setting is the Sobolev space W^1,1(0,1). When α=0 the Inf is achieved in W^1,1(0,1). However, when α =1, minimizers need not exist in W^1,1 (0,1). One is led to introduce a relaxed functional defined on the space BV(0,1), whose minimizers always exist and can be viewed as generalized solutions of the original ill-posed problem.