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I am currently studying GMT and a topic that has popped up in the course is the use of calibrations as a tool for proving that a particular set $E$ attains the minimum for the problem $$min \left \lbr …

My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that $$ \int_\Omega |f|^p (x) w(x) dx \text …

It is well known that given a $n \times n$ matrix $A$, it holds that $$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$ I would need a full representation of $ \de …

Let $\Omega$ be a ball of radius $r$. It is well known that given a function $f \in W^{1,p} (\Omega)$, it holds the Poincaré-Wirtinger inequality $$ \left(\int_\Omega |f - f_\Omega|^p dx \right)^{1/p} …

I've just started learning about some methods to deal with parabolic equations, and in a lot of papers they refer to the book "Quelques méthodes de résolution des problèmes aux limites non linéaires" …

Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \to 0 …

It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as $$ F= \nabla V+ \nabla \times R$$ with $V$ a potential and $R$ another vector field. These components …

Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$ …

I would need to get acquainted to the subject of stochastic Stokes flows, so studying Stokes equations under some noise of some kind, let's say an additive white noise to begin with. The problem is th …