# Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.

4,778 questions
Filter by
Sorted by
Tagged with
23 views

• 804
43 views

• 523
1 vote
25 views

### Sobolev inequality for cubes

Consider the fallowing result from Evans, 2010, page 279: Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then ...
92 views

### Is it possible to find the maximum of $r\ln\left(\ln\left(1+\frac{1}{r}\right)\right)$ for $r\in(0,1)$?

I was working on Chapter 5, Problem 14 of Evan's Partial Differential Equations 2nd ed and to prove integrability of $\ln \left (\ln \left (1+\dfrac{1}{|x|}\right )\right )$ on the unit $n$-ball, ...
• 51
16 views

### What is the relationship between weak derivatives and the decay of the coefficients in the eigenfunction expansion?

Suppose we have a function $f: [0,1] \to \mathbb{R}$ that has $2\beta$, $\beta \in \mathbb{N}$, weak derivatives. Defining $e_k(x) := \sqrt{2}\sin(k \pi x)$ the orthonormal sine-basis of $L^2([0,1])$, ...
45 views

### Relationship between Sobolev-Slobodeckij spaces and Besov spaces

I am trying to understand these two different ways of defining fractional Sobolev spaces. In particular, I want to determine embeddings or equality between the Besov spaces $B^{s}_{p,p}$ and the ...
• 5,402
41 views

### Weak derivative of power function

Let $f\in {W}^{1,p}(0,1)$ ($1\leq p\leq\infty$) such that $f>0$ a.e., and let $g = f^\alpha$ ($0<\alpha<1$). I am wondering if $$g'=\alpha f^{\alpha-1} f'\ (weak)\qquad a.e. ?$$ Can somebody ...
26 views

37 views

### About the continuity of the integral on the boundary of a ball of a $H^1$ function

I’m considering a $H^1$ function u on a open domain D. Is the integral: $$\int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$ continuous with respect to x? I tried to prove that it’s differential by ...
62 views

### Assumptions in Schauder Fixed Point Theorem

I have a - maybe slightly stupid - question about the Schauder-Fixed-Point Theorem. The formulation I have in mind is: Let $A$ be a closed, convex, nonempty subset in a Banach space $(X,\|\cdot\|)$, ...
• 158