Search Results

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: …

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 …

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^ …

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 …

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 $0$ …

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 …

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 …

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 …

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 …

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