Is there some inequality which relates ∫ Ω | f | 2 and ∫ ∂ Ω | f | 2 for some f ∈ L 2 ( Ω ) ∩ L 2 ( ∂ Ω )?

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Is there some inequality which relates      ∫          Ω            |    f            |              2             and         ∫          ∂      Ω            |    f            |              2      for some   f  ∈      L          2        (  Ω  )  ∩      L          2        (  ∂  Ω  )?

lywiau63 · Accepted Answer

Let   Ω be the unit disk in             R        2  . The function   f  (  x  ,  y  )  =  (  1  −      x    2    −      y    2        )          1              /            2       satisfies      ∫    Ω        |    f            |        2    =      π    2      and        ∫          ∂      Ω            |    f            |        2    =  0.On the other hand the functions       f    n    (  x  ,  y  )  =  (      x    2    +      y    2        )          n              /            2       satisfy      ∫    Ω        |        f    n              |        2    =      π          n      +      1          and        ∫          ∂      Ω            |        f    n              |        2    =  2  πso that in general the ratio                    ∫        Ω                    |            f                        |                2                            ∫                  ∂          Ω                            |            f                        |                2            is bounded neither above nor below.

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