# Tagged Questions

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

58 views

### Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. ...
40 views

### Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
29 views

### Are compactly supported functions in $W^{1,2}(\mathbb R^n)$ also in $W_0^{1,2}(\mathbb R^n)$? See for proof?

The question is stated clearly in the title. On the one hand, it seems obvious (and I give an argument below). On the other hand, after a quick search I haven't been able to find the statement ...
37 views

28 views

### Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions

The page where the proof is is on Google Books. A picture of the statement and the proof. I reproduce the statement of the result: Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. ...
53 views

### How is the $H^{1/2}$ norm of function defined on a subset of the boundary?

Let $\Omega\subset \Omega^d$, $d\in \{2,3\}$, be a bounded $d$-polyhedron with $n$ faces. Denote the faces of $\partial\Omega$ as $\{e_i\}_{i=1}^n$. Let $u\in H^{1/2}(\partial\Omega)$ Taking the ...
28 views

### Density of smooth positive functions

Let $\Omega$ be an open bounded set of $R^n$. For $f\in L^2(\Omega)$ such that $f>0$, a.e. in $\Omega,$ there is $(f_k)\subset W^{2,\infty}(\Omega)$ such that $f_k\to f$ in $L^2(\Omega)$. My ...
19 views

24 views

### Integral of a weak derivative

While reading chapter 6 of John Hunter's notes (https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf) I got stuck on some steps. I think they are all based on a similar idea as the following. Let ...
19 views

### Compactness Sobolev embedding for even functions on $\mathbb{R}$.

It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial ...
32 views

### Understanding multiindex notation and the Sobolev Space $W^{1,p}$.

The notation comes from Evans Partial Differential Equations. From Appendix A, we are given information about multiindex notation. Assume $u : U \rightarrow R$, $x \in U$. (a) A vector of the ...
51 views

### Find $v \in H^1(0,1)$ which satisfies the following

Determine the function $v \in H^1(0,1)$ which satisfies the equation $u(0)=\langle u,v \rangle_{H^1}$ for all $u\in H^1(0,1)$ . It is clear that on $H^1(0,1)$; $u(0)=\int_{0}^{1}(uv+u'v')$.What can ...
41 views

### Mistake in a PDE book regarding Lebesgue's differentiation theorem? To do with weak formulation

I'm reading "Elliptic and Parabolic Equations" by Wu, Yin and Wang. In Section 4.2, they consider the heat equation given $u_0 \in L^\infty$ and $f \in L^\infty$ $$u_t - \Delta u = f$$ $$u(0) = u_0$$ ...
34 views

### Why the need of Sobolev spaces in this proof of isoperimetric inequality?

I was reading the chapter about isoperimetric inequalities in DaCorogna's book "Introduction to The Calculus of Variations". The isoperimetric inequality is proved to be equivalent to Wirtinger ...
52 views

### If an $H^1$ function vanishes on a set of positive measure, its $L^2$ norm is controlled by the gradient

I am trying to solve question 15 from Evans' PDE book, chapter 5. You have a set of positive measure, subset of the unit ball $B$, such that $u$ is equal to zero on that set. Then, one can show that: ...
71 views

61 views

### Definition of weak time derivative

My quesion involves the weak time derivative. In the book: 'Partial Differential Equations' by Evans the time derivative $u'$ of a function $u: [0,T] \rightarrow H^1_0(U)$ is defined by an element ...
29 views

### The eigenvalue for mollified function

Let $u_k$ be eigenfunctions for operator $-\Delta$ over domain $\Omega\subset \mathbb R^2$, open bounded, smooth boundary. Then, we know that $E:=\{u_k\}_{k=1}^\infty$ forms a basis for $L^2$ and we ...
29 views

### Operators of order $k$ between Sobolev spaces

It may sound like tautology, but I have a problem in proving that differential operator of order $k$ is an operator of order $k$. What exactly I'm asking for? Let $M$ be a manifold and $E,F$ be two ...
20 views

### How to show that a function is in a Sobolev space

This question is about the solution of exercise 1.20 in Elman, Silvester, Wathen. Finite Elements and Fast Iterative Solvers. (The first Chapter of the book is open access and available, for example, ...
50 views

40 views

### What is the square root of the Laplace operator?

Let $\Delta$ be the Laplace operator $$\Delta f = \sum_{i=1}^d \frac{\partial^2 f}{\partial x^2_i}$$ with $Dom(\Lambda) = H^1_0(\mathcal{O}) \cap H^2(\mathcal{O})$ where ...
19 views

37 views

### $L^{2}$ convergence of sequence $|u_{j}|^{p}\nabla u_{j}$

Suppose a sequence $\{u_{j}\}$ is bounded in the Sobolev space $H^{2+\epsilon}(\Omega)$, for $\epsilon>0$, where $\Omega$ is say a bounded, $C^{\infty}$ domain. Here, the fractional space ...
### Product of $W^{1,p}_0$ functions
Let $p>n$, and let $f,g\in W^{1,p}_0(\mathbb{R}^n)$ be two sobolev functions. Prove that $fg\in W^{1,p}_0(\mathbb{R}^n)$. I was able to prove the Leibniz formula for weak derivativatives, but ...
How the weak derivative of the sign function \begin{equation*} sign(x)=\left\{ \begin{array}{rl}1 & \text{if } x> 0,\\ 0 & \text{if } x=0, \\ -1 & \text{if } x<0 \\ ...