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Unlike their Hölder cousins, most Sobolev spaces are reflexive Banach spaces. Reflexivity is a highly desirable feature for variational problems because it gives a little bit of compactness, enough …

There is a coordinate free way of defining Sobolev spaces of sections of a vector bundle $E$ over a manifold $M$. You need to make a few choices: a metric $g$ on $M$, a metric $h$ on $E$ and a con …

$\DeclareMathOperator{\dist}{dist}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\pa}{\partial}$ Suppose that $N=2$ and $\Omega$ is is the unit disk. Choose $$ u= -1+ar^4+br^5\in C^2(\overline{\Omeg …

Note first of all that $H^{-1}$ will contain objects that are not functions, such as the Dirac $\delta$ concentrated at a point. (Think that $2\delta_0=\frac{d^2}{dx^2}|x|$.) The correct definitio …

I'll start with several known facts. Proposition 1. Suppose that $E$ is a real Hilbert space with norm $\Vert -\Vert$ and $K: E\to E$ is a a compact, selfadjoint positive operator. Denote …

$J$ is not coercive in $W^{1,2}$ For that to happen you need to show that $\Vert u_n\Vert_{1,2}\to \infty$ implies $J(u_n)\to \infty$. Take for example the function $u_n$ which is identically $0$ …

Clearly, what you call $\newcommand{\bn}{\boldsymbol{n}}$ $\nabla u\cdot \bn $ is the normal derivative $\frac{\partial u}{\partial \bn}$. The trace theorem (see e.g. Lions and Magenes, Non-Homogene …

Here is a classical theorem. $\newcommand{\bR}{\mathbb{R}}$ Suppose that for $0< a,b<\infty$ $$\sup_x\left(\int_{\bR^n} |k(x,y)|^a dy\right)^{\frac{1}{a}}=M_1(k)<\infty, $$ $$\sup_y\left(\int_{\bR^ …

The answer is yes, with the caveat indicated in the comment below. Consider an arbitrary extension $\hat{\theta}\in H^2(\Omega)$. Now choose a compactly supported smooth function $\eta$ such that …

Sobolev's original proof is different from the two approaches described by Deane. He uses certain integral formulae. You can read about this approach in the classic monograph C. Morrey: …

If you read French then this book is the place you are looking for Brézis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holl …