On the geometry of the pp-Laplacian operator

The $p$-Laplacian operator $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $\Omega$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We report also on results concerning the normalized or game-theoretic $p$-Laplacian $$\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$$ and its parabolic counterpart $u_t-\Delta_p^N u=0$. These equations are homogeneous of degree 1 and $\Delta_p^N$ is uniformly elliptic for any $p\in (1,\infty)$. In this respect it is more benign than the $p$-Laplacian, but it is not of divergence type.Comment: 15 pages, 5 figures, Survey lecture given at the WIAS conference "Theory and Applications of Partial Differential Equations" in Dec. 201

Overdetermined boundary value problems for the ∞\infty-Laplacian

We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $\Omega$ of $\R^n$ and discuss what kind of implications on the geometry of $\Omega$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.Comment: 9 pages, 1 figur

Abstraction and registration: conceptual innovations and supply effects in Prussian and British Copyright (1820-50)

It is one of the orthodoxies of modern copyright law that the enjoyment and the exercise of the rights granted “shall not be subject to any formality” (Berne Convention 1886, Berlin revision 1908, Art.4), such as a registration requirement. In this article, we trace the origins of this provision to a conceptual shift that took place during the early 1800s. Specific regulations of the book trade were superseded by the protection of all instantiations (such as performances, translations and adaptations) of abstract authored work. For two seminal copyright acts of the period, the Prussian Act of 1837 and the UK Act of 1842, we show there was considerable concern about the economic implications of this new justificatory paradigm, reflected in a period of experimentation with sophisticated registration requirements. We indicate market responses to these requirements and plea for a reconsideration of “formalities” as redressing justificatory problems of copyright in the digital environment

Johann Gottlieb Fichte, and the Trap of Inhalt (Content) and Form: An Information Perspective on Music Copyright

In the digital environment, copyright law has become trapped in an assessment of what has been taken, rather than what has been done with copied materials and elements. This expands the scope of copyright into areas where it should not find infringement (such as sampling, mash-ups and other transformative uses) while encouraging activities that are problematic (such as hiding sources). This article argues that the trap was laid by the German idealist philosopher Johann Gottlieb Fichte whose influential 1793 article Proof of the Unlawfulness of Reprinting for the first time distinguishes Inhalt (i.e. content free to all) and Form (i.e. the author’s inalienable expression) as copyright categories. It is shown that Fichte’s structure conflates norms of communication and norms of transaction. An alternative path for copyright law in an information society is sketched from a separation of these norms: copying should be assessed from (i) the attribution of sources, and (ii) the degree to which original and derivative materials compete with each other. Throughout the article, transformative practices in music set the scene

Droplet condensation and isoperimetric towers

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized

The problem of minimal resistance for functions and domains

Here we solve the problem posed by Comte and Lachand-Robert in [SIAM J. Math. Anal., 34 (2002), pp. 101–120]. Take a bounded domain Ω ⊂ R2 and a piecewise smooth nonpositive function u : ¯Ω → R vanishing on ∂Ω. Consider a flow of point particles falling vertically down and reflected elastically from the graph of u. It is assumed that each particle is reflected no more than once (no multiple reflections are allowed); then the resistance of the graph to the flow is expressed as R(u; Ω) = 1 |Ω| Ω(1 + |∇u(x)|2)−1dx. We need to find infΩ,u R(u;Ω). One can easily see that |∇u(x)| 1/2. We prove that the infimum of R is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, and partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem [Amer. Math. Monthly, 70 (1963), pp. 697–706]