Intereting Posts

Continuous function on closed unit ball
Which of these numbers is the biggest
conversion of laplacian from cartesian to spherical coordinates
Automorphism group of the general affine group of the affine line over a finite field?
Demonstration: If all vectors of $V$ are eigenvectors of $T$, then there is one $\lambda$ such that $T(v) = \lambda v$ for all $v \in V$.
Gradient steepest direction and normal to surface?
Principal Bundles, Chern Classes, and Abelian Instantons
Relationship between intersection and compositum of fields
Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$
Fourier transform of $\text{sinc}^3 {\pi t}$
Growth-rate vs totality
What's the lower bound of the sum $S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n)$?
Units and Nilpotents
Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$?
surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem:

$$

\Delta \omega = 0 \text{ in } M \\

\omega = f \text{ in } \partial M

$$

I would like to find some bounds on $ \omega, \partial_x \omega, \partial_x^2 \omega, \partial_y \omega, \partial_y^2 \omega $. Well, | $ \omega $ | is easy. But, is there anything I can say about the derivatives, in terms of $ f $?

If one considers the case where $ M $ is a square, then the usual fourier tricks are helpful. Is there something I can do for general regions $ M $?

- Formula for Sum of Logarithms $\ln(n)^m$
- Physicists, not mathematicians, can multiply both sides with $dx$ - why?
- Extension of bounded convex function to boundary
- Recurrent points and rotation number
- How to prove that the set of rational numbers are countable?
- Uniqueness existence of solutions local analytical for a PDE

- Integral Representation of Infinite series
- Why are additional constraint and penalty term equivalent in ridge regression?
- Continuity of the derivative at a point given certain hypotheses
- Analog of $(a+b)^2 \leq 2(a^2 + b^2)$
- Addition is to Integration as Multiplication is to ______
- Prove the uniqueness of poisson equation with robin boundary condition
- Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $
- How to prove that a set R\Z is open
- Closed form for definite integral involving Erf and Gaussian?
- Why is the Daniell integral not so popular?

As pointed out in comments by Shucao Cao and Ray Yang, Gilbarg-Trudinger is the place to look up such estimates: it does not have `3704`

MathSciNet citations for nothing (as of today). In addition to global Hölder and Sobolev estimates (which require appropriate smoothness of $\partial M$ and $f$) there is a simple *interior* estimate for the Laplace equation, which applies in great generality. I quote it below.

**Theorem 2.10.** Let $u$ be harmonic in $\Omega\subseteq \mathbb R^n$ and let $\Omega’$ be a compact subset of $\Omega$. Then for any multi-index $\alpha$ we have

$$\sup_{\Omega’} |D^\alpha u|\le \left(\frac{n|\alpha|}{d}\right)^{|\alpha|}\sup_\Omega |u|$$ where $d=\operatorname{dist}(\Omega’,\partial\Omega)$.

By the maximum principle, you can replace $\sup_\Omega |u|$ with $\sup_{\partial \Omega}|f|$ where $f$ is the boundary data.

- Who is buried in Weierstrass' tomb?
- Is it coherent to extend $\mathbb{R}$ with a reciprocal of $0$?
- What are some good Fourier analysis books?
- Convergence in weak topology implies convergence in norm topology
- A problem For the boundary value problem, $y''+\lambda y=0$, $y(-π)=y(π)$ , $y’(-π)=y’(π)$
- Baby Rudin vs. Abbott
- Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?
- $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample
- Question on partial differential equation
- What's wrong with this argument? (Limits)
- “Any finite poset has a maximal element.” How to formalize the proof that is given?
- Intuitive/heuristic explanation of Polya's urn
- Is $e^x=\exp(x)$ and why?
- Evaluate $\int \frac 1{x^{12}+1} \, dx$
- Andrei flips a coin over and over again until he gets a tail followed by a head, then he quits. What is the expected number of coin flips?