Let CR be a bounded, open subset with smooth boundary an. Consider the boundary-value problem (BVP)...

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Let CR" be a bounded, open subset with smooth boundary an. Consider the boundary-value problem (BVP) -Au = f u=0 where fEL²(2). Further define the functional in Ω on an 1 [w] := √ ( 1 || VW|| ² -- wf ) dx Vw on the class of functions I = {w = H (2): w≥g on an} where 8: given smooth function, called the obstacle. Consider the following minimization problem: (MIN) I(u) := min1[w]. wer → R is a (a) Show that if u Ersatisfies (MIN), then u solves (BVP). (b) Show that there exists a function er satisfying (MIN) (c) Show that the function ũ appearing in (b) is the unique solution to the mini- mizing problem (MIN). (d) Conclude that ũ is the unique solution to (BVP). [Be careful here] Let CR" be a bounded, open subset with smooth boundary an. Consider the boundary-value problem (BVP) -Au = f u=0 where fEL²(2). Further define the functional in Ω on an 1 [w] := √ ( 1 || VW|| ² -- wf ) dx Vw on the class of functions I = {w = H (2): w≥g on an} where 8: given smooth function, called the obstacle. Consider the following minimization problem: (MIN) I(u) := min1[w]. wer → R is a (a) Show that if u Ersatisfies (MIN), then u solves (BVP). (b) Show that there exists a function er satisfying (MIN) (c) Show that the function ũ appearing in (b) is the unique solution to the mini- mizing problem (MIN). (d) Conclude that ũ is the unique solution to (BVP). [Be careful here]