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This paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functionalJ  ⁢   (  k  )    :=    (   1  /  2   )   ⁢    ∥    u  ⁢   (  0  ,  ⋅  ;  k  )    -  f   ∥L  2   ⁢   (  0  ,  T  )    2{J(k):=(1/2)\lVert u(0,\cdot\,;k)-f\rVert^{2}_{L^{2}(0,T)}}    corresponding to an inverse coefficient problem for the1  ⁢  D{1D}    parabolic equationu  t   =    (   k  ⁢   (  x  )   ⁢   u  x    )   x{u_{t}=(k(x)u_{x})_{x}}    with the Neumann boundary conditions-   k  ⁢   (  0  )   ⁢   u  x   ⁢   (  0  ,  t  )=   g  ⁢   (  t  ){-k(0)u_{x}(0,t)=g(t)}    andu  x   ⁢   (  l  ,  t  )    =  0{u_{x}(l,t)=0}   . In addition, compactness and Lipschitz continuity of the input-output operator

Lipschitz continuity of the Fréchet gradient in an inverse coefficient problem for a parabolic equation with Dirichlet measured output