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$W^{-k,p}$ is the dual of $W_0^{k,q}$ if $p>1$. For $p=1$ this is not a natural definition. You should use the alternative definition that $W^{-1,1}$ is the set of all distributions of the form $f_0+\ …

Consider the operator $d^{2m}/dx^{2m}$ with natural boundary conditions $u^{(m)}=u^{(m+1)}=...=u^{(2m-1)}=0$. This operator is self-adjoint in $L^2$, and $W_2^m$ is its form domain. Hence the eigenfun …

No, something like this is not true. In any negative Sobolev space, you can find a function f with a point singularity such that $f^2$ is not integrable. Hence $f^2$ is not even a distribution, and ce …

This is not a full answer, but it shows you can do better than p=6. In the following, subscripts x and y refer to the x and y dependence. You have $$u \in H^{2/3}_x ( L^2_y )\cap L^2_x( H^1_y ).$$ By …

The abstract of the following paper sounds like it might be relevant to your first question. Abstract. We prove that the well-known trace theorem for weighted Sobolev spaces holds true under min …

Either type of convergence implies distributional convergence, among other things. So the limits must be the same.

In general your problem does not have a solution. Suppose f has zero average and one simple zero $x_0$ in the interval. If $y$ is to be smooth with zero average, then it must also have a simple zero a …

Tooting my own horn: You can start with Renardy and Rogers, An Introduction to Partial Differential Equations. You will find references to the original papers there.

Yes, it is true. As you already found, it suffices to show that $H^2$ functions are multipliers in $H^{1/2}$. This follows by interpolation, since it is easy to show that $H^2$ functions are multiplie …

$H^1(R^{n+1}_+)$ embeds into $C(R^+,L^2(R^n))$, and you can apply $\gamma$ there. So it is correct. Of course this does not imply that $\gamma$ maps $H^{1/2}(R^n)$ to itself.

The question is whether a function in $H^1(\Omega)\cap C(\bar\Omega)$ which vanishes on $\partial\Omega$ is in fact in $H^1_0(\Omega)$. This is true. A more general result is proved in D. Swanson and …

Yes, this extension works. Your extended function is clearly in $H^1$ on the annulus as well as in $R^n\backslash B_2$. Since the traces on $\partial B_2$ agree, it is then in $H^1$ on $R^n\backslash …

If $\Omega$ is a half space, you can construct w by Fourier transforms. Any proof of the inverse trace theorem in the literature will show you specifics. For general $\Omega$, you use partition of uni …

Let $\Omega=\{(x,y)\,|\,0<x<1,0<y<w(x)\}$, where $w(x)\to 0$ as $x\to 0$. Let $\phi$ be any smooth positive function on $[0,1]$, which vanishes at the end points. Now define $$\psi(x,y)=(1-x)\phi(y/w( …

For the moment, fix $$c=\int_0^1 yf(x,y)\,dy.$$ It is clear that we must have $0\le c\le 1$ to satisfy the constraints on $f$. We can write the integral to be minimized as $$\int_0^1\Bigl(x^2-2cx+\int …