Intereting Posts

Limit involving incomplete gamma function
Number of triangles inside given n-gon?
Number of nonnegative integer solutions to $x_1+x_2+x_3\le10$ with $x_1 \ge 1\ ,\ x_2\ge3$
If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $$ is divisible by 56 but not 49.
How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution
Extending a morphism of schemes
Trick to find multiples mentally
Text on Group Theory and Graphs
Evaluate the series $\sum_{n=1}^{\infty} \frac{2^{}+2^{-}}{2^n}$
Integers expressible in the form $x^2 + 3y^2$
Can a function have a strict local extremum at each point?
Integration Techniques – Adding values to the numerator.
$x^3+48=y^4$ does not have integer (?) solutions
Converse of interchanging order for derivatives
Root of multiplicity?

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 $?

- Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$
- This integral is defined ? $\displaystyle\int_0^0\frac 1x\:dx$
- Solving a challenging differential equation
- Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$
- Showing a function is not uniformly continuous
- $f$ is a real function and it is $\alpha$-Holder continuous with $\alpha>1$. Is $f$ constant?

- Continuous increasing bounded function, derivative
- Does absolute convergence of a sum imply uniform convergence?
- A functional relation which is satisfied by $\cos x$ and $\sin x$
- Real and imaginary parts of a complex-valued function
- For $b \gt 2$ , verify that $\sum_{n=1}^{\infty}\frac{n!}{b(b+1)…(b+n-1)}=\frac{1}{b-2}$.
- p-seminorms on smooth functions are equivalent
- The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle
- Fundamental Theorem of Calculus for distributions.
- A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function
- Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$

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.

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- Explicit descriptions of groups of order 45
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- Simplify the fraction with radicals
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- What could the ratio of two sides of a triangle possibly have to do with exponential functions?
- variance inequality