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  1. I am a bit unsure about the role of the trace operator. I understand that if you have a PDE that is solved by a function $u$ in some Sobolev space, then it's not necessarily defined on the boundary since $u$ is in $L^p$ and so can be "redefined" on the boundary as it's a null set. But sometimes we have boundary conditions that $u$ needs to satisfy so that uniquely fixes $u$. If we don't have such boundary conditions, why does it matter what $u$ is on the boundary? What role does the trace operator play?

  2. If $u$ is in $H^{r+2}(\Omega)$ for all $r$ where $\Omega$ is compact, then $u$ is also in $C^k(\overline{\Omega})$ for all $k$. This is a fact but I don't know why? How does existence and integrability of $u$ and its derivatives imply continuity of $u$ and its derivatives?

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Too tired now to explain in detail. Maybe I'll do so tomorrow if no one else does. In the mean while, I addressed both issues in my lecture notes. The answer to the first question, in the context of PDEs, basically boils down to a priori estimates. The second question is answered by Morrey's inequalities. Another good reference that addresses the issues is Adams' Sobolev spaces. –  Willie Wong Dec 4 '11 at 22:06
    
To give one version of a quick summary to how trace theorems work: you define on $W^{k,p}\cap C$ (continuous functions) which is dense in $W^{k,p}$. The estimates show that the mapping into $L^q$ is bounded and linear. So by the BLT theorem there is a unique continuous extension to the whole of $W^{k,p}$. –  Willie Wong Dec 4 '11 at 22:12
    
Thanks. I don't suppose the rest of the CCA Analysis lecture notes are available for public consumption? –  Court Dec 7 '11 at 17:14
    
AFAIK the course websites are only available for CCA students. But individuals may have the notes on their websites. –  Willie Wong Dec 8 '11 at 8:58

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As Willie explained the intuition of the definition for the trace operator, maybe I can say some words about the boundary condition in Differential Equations, and hope that it can help. Take the elliptic equation as example, if we want to consider the shape of a membrane under outer force, we are lead to equation like $-\Delta u=f$, assume the boundary of the membrane is good enough and is fixed (actually now the boundary function is the trace of the shape function), then this is the Cauchy problem for the equation.

Moreover, because in elliptic equation we have good enough a priori estimate(use Newton potential we could see what the solution looks like), we can solve this Cauchy problem for general boundary data(which might not be continuous, for example $L^p$), in this case the solution is not continuous up to the boundary, but converge in $L^p$ sense, which is similar to the definition of trace operator.

So for me, "trace" (generalization of the boundary value problem)just makes the problem well-posed.

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Thanks for the reply. –  Court Dec 7 '11 at 17:14

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