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Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with norm $|u|=(\|u\|^2 +\|\Delta u\|^2)^{\frac{1}{2}}$, where $\|\cdot\|$ is the norm of $L^2(\Omega).$

I want to explicit the adjoint operator $B^*$ of $B$ in $(H,|.|)$.

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This seems more appropriate for math.stackexchange.com. But note that it suffices to find the adjoint for compactly smooth functions. –  Deane Yang May 5 '13 at 15:26
    
Cross-post at MSE: math.stackexchange.com/questions/382602/… –  gerw May 6 '13 at 7:49
    
Yes he is a co-worker –  reseacher May 6 '13 at 13:26

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