摘要
We explore minimization problems of the form ■ where u is a function defined on(0, 1),(ai) are k given points in(0, 1), with k ≥ 2,( fi)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 W1,1(0, 1). When α = 0 the Inf is achieved in W1,1(0, 1). However, when α = 1, minimizers need not exist in W1,1(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.
