Flatness of Bernoulli jets
For \(m \in (0, +\infty),\) \( p \in (1, +\infty)\) and a domain \( \Omega \subseteq \mathbb{R}^n\), consider the following functional on \(W^{1, p}(\Omega)\) as \[ \mathcal{F}_\Omega(u) = \int_\Omega \frac{\mid\nabla u(x)\mid^p}{p} + m_{\chi_{(-1, 1)}} u(x) dx. \] In this paper, the author mainly deals with minimizers of the above functional. An element \( u \in W^{1, p}(\Omega)\) is a local minimizer for \(\mathcal{F}\) in \(\Omega\) if \( \mathcal{F}_\Omega(u) \leq \mathcal{F}_\Omega(u + \varphi),\) for any \(\varphi \in C_0^{\infty}(\Omega).\) If \(u\) is defined in the whole space and it is a local minimizer in any domain, then it is said that \(u\) is a Class A minimizer. The author shows that the level sets of minimizers of a possibly singular or degenerate elliptic variational problem related with fluid jets possess a Harnack-type inequality. In fact, it is shown that if the zero level set of a minimizer is trapped in a rectangle whose height is small enough, then in a smaller neighborhood, it can be trapped in a rectangle with even smaller height.
scientific article; zbMATH DE number 5076561
scientific article