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Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality $$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla …

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for eve …

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and $$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} …

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$. We say that $\sigma$ has weak divergence if there exists …

It is well known that for Lipschitz domains $\Omega\subset \Bbb{R}^2$ one can define linear extension operators $$ E: H^1(\Omega) \to H^1(\Bbb{R}^2).$$ I am interested in explicit upper bounds for the …