Intereting Posts

Are there real world applications of finite group theory?
“This statement is false.”
Why is the endomorphism ring of $\mathbb{Z}\times\mathbb{Z}$ noncommutative?
Want less brutish proof: if $a+b+c=3abc$ then $\frac1a+\frac1b+\frac1c\geq 3$
Taylor series for different points… how do they look?
Sum of the following series upto n terms:$\sum_{k=1}^n \frac {k}{(k+1)(k+2)} 2^k$
If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose center is $C$ be such that $CP$ is perpendicular to $CQ$
Newton-Raphson for reciprocal square root
Find $\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+…}}}}$
Limit distribution of infinite sum of Bernoulli random variables
Projective Co-ordinate Geometry
modular form -Petersson inner product
$SL(n)$ is a differentiable manifold
Fun Linear Algebra Problems
Does “This is a lie” prove the insufficiency of binary logic?

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$.

I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to take this further and show that $W^{1,p}(\Omega) \Subset L^{\infty}(\Omega)$ where $\Omega \subset \mathbb{R}^{n}$ is $C^{1}$?

- Example of converging subnet, when there is no converging subsequence
- A Banach space is reflexive if and only if its dual is reflexive
- Graph of symmetric linear map is closed
- Continuity of infinite $l_p$ matrix
- Reference request: Fourier and Fourier-Stieltjes algebras
- Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

- Distance from a point to a plane in normed spaces
- Weak convergence of a sequence of characteristic functions
- Double dual of the space $C$
- Norm for pointwise convergence
- Analyticity: Uniform Limit
- Diophantine number has full measure but is meager
- Perturbation by bounded operators
- Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?
- What do we need Sobolev-spaces $W^{k, p}$ with $p \neq 2$ for?
- Bounded inverse operator

Here is a direct proof for $L^\infty$, without Hölder spaces or Morrey inequality.

- By the Arzelà–Ascoli theorem, every infinite bounded subset of $W^{1,\infty}$ has a limit point in $L^\infty$.
- Therefore, the embedding of $W^{1,\infty}$ into $L^\infty$ is compact.
- A compact operator maps weakly convergent sequences into convergent sequences.

*Old answer*: Every $C^1$ domain, and more generally a Lipschitz domain, is a *Sobolev extension domain*, meaning that Sobolev functions on it can be extended to Sobolev functions on $\mathbb R^n$. In particular, all embedding theorems for Sobolev spaces hold on such domains.

When $p>n$, Morrey’s inequality gives a continuous embedding of $W^{1,p}$ into $C^\beta$ with $\beta=1-n/p$. In turn, $C^\beta$ compactly embeds into $C^\alpha$ for $0<\alpha<\beta$. See Is there a reference for compact imbedding of Hölder space? This topic was also discussed in Compact, continuous embeddings of $H^s := W^{s,2} \leftrightarrow C^{(\alpha)}$ (in one dimension).

Under your assumptions, you can get $u_m\to u$ in any space $C^\alpha$ with $\alpha\in (0,1)$. *A fortiori*, $u_m\to u$ in $L^\infty$.

- Closed form of arctanlog series
- Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?
- Big-O Notation – Prove that $n^2 + 2n + 3$ is $\mathcal O(n^2)$
- Prove that the difference between two rational numbers is rational
- Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$
- Prove if $f: \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
- Derivative of convolution
- Eigenfunction of the Fourier transform
- Christoffel Symbols as Tensors
- $$ is not true or false
- Finding the value of $k$ for parallel/orthogonal planes
- Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$
- Integration of $e^{ax}\cos bx$ and $e^{ax}\sin bx$
- Uniqueness existence of solutions local analytical for a PDE
- Are those two numbers transcendental?