Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

Liouville theorems for stable solutions of biharmonic problem

We prove some Liouville type results for stable solutions to the biharmonic problem $\Delta ^2 u= u^q, \,u>0$ in $\mathbb{R }^n$ where $1 < q < \infty $ . For example, for $n \ge 5$ , we show that there are no stable classical solution in $\mathbb{R }^n$ when $\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}$ .     

Oka properties of ball complements

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension ${<}n$ to the complement $\mathbb {C}^n{\setminus } L$ of a compact convex set $L\subset \mathbb {C}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X \le n$ . We also construct proper holomorphic maps, immersions and embeddings $X\rightarrow \mathbb {C}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.     

An illumination problem: optimal apex and optimal orientation for a cone of light

Let $\{a_i:i\in I\}$ be a finite set in $\mathbb R ^n$ . The illumination problem addressed in this work is about selecting an apex $z$ in a prescribed set $Z\subseteq \mathbb R ^n$ and a unit vector $y\in \mathbb R ^n$ so that the conic light beam $$\begin{aligned} C(z,y,s):= \{x \in \mathbb R ^n : s\,\Vert x-z\Vert - \langle y, x-z\rangle \le 0\} \end{aligned}$$ captures every $a_i$ and, at the same time, it has a sharpness coefficient $ s\in [0,1]$ as large as possible.     

The distribution of the zeros of the Goss zeta-function for A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)

Let $F$ be a global function field over a finite constant field and $\infty $ a place of $F$ . The ring $A$ of functions regular away from $\infty $ in $F$ is a Dedekind domain. For such $A$ Goss defined a $\zeta $ -function which is a continuous function from $\mathbb{Z }_p$ to the ring of entire power series with coefficients in the completion $F_\infty $ of $F$ at $\infty $ . He asks what one can say about the distribution of the zeros of the entire function at any parameter of $\mathbb{Z }_p$ . In the simplest case $A$ is the polynomial ring in one variable over a finite field. Here the question was settled completely by J. Sheats, after previous work by J. Diaz-Vargas, B. Poonen and D. Wan: for any parameter in $\mathbb{Z }_p$ the zeros of the power series have pairwise different valuations and they lie in  $F_\infty $ . In the present article we completely determine the distribution of zeros for the simplest case different from polynomial rings, namely $A=\mathbb{F }\,\!{}_2[x,y]/(y^2+y+x^3+x+1)$ —this $A$ has class number $1$ , it is the affine coordinate ring of a supersingular elliptic curve and the place $\infty $ is $\mathbb{F }\,\!{}_2$ -rational. The answer is slightly different from the above case of polynomial rings. For arbitrary $A$ such that $\infty $ is a rational place of $F$ , we describe a pattern in the distribution of zeros which we observed in some computational experiments. Finally, we present some precise conjectures on the fields of rationality of these zeroes for one particular hyperelliptic $A$ of genus  $2$ .     

Convergence of the eigenvalues of the p-Laplace operator as p goes to 1

Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ .     

p-Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes

In this paper we study, for given $p,~1<p<\infty $ , the boundary behaviour of non-negative $p$ -harmonic functions in the Heisenberg group $\mathbb{H }^n$ , i.e., we consider weak solutions to the non-linear and potentially degenerate partial differential equation $$\begin{aligned} \sum _{i=1}^{2n}X_i(|Xu|^{p-2}\,X_i u)=0 \end{aligned}$$ where the vector fields $X_1,\ldots ,X_{2n}$ form a basis for the space of left-invariant vector fields on $\mathbb{H }^n$ . In particular, we introduce a set of domains $\Omega \subset \mathbb{H }^n$ which we refer to as domains well-approximated by non-characteristic hyperplanes and in $\Omega $ we prove, for $2\le p<\infty $ , the boundary Harnack inequality as well as the Hölder continuity for ratios of positive $p$ -harmonic functions vanishing on a portion of $\partial \Omega $ .     

Subsets of full measure in a generic submanifold in \mathbb C ^n are non-plurithin

In this paper we prove that if $I\subset M $ is a subset of measure $0$ in a $C^2$ -smooth generic submanifold $M \subset \mathbb C ^n$ , then $M \setminus I$ is non-plurithin at each point of $M$ in $\mathbb C ^n$ . This result improves a previous result of A. Edigarian and J. Wiegerinck who considered the case where $I$ is pluripolar set contained in a $C^1$ -smooth generic submanifold $M \subset \mathbb C ^n$ (Edigarian and Wiegernick in Math. Z. 266(2):393–398, 2010). The proof of our result is essentially different.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Coset intersection of irreducible plane cubics

In a projective plane $PG(2,\mathbb K )$ over an algebraically closed field $\mathbb K $ of characteristic $p\ge 0$ , let $\Omega $ be a pointset of size $n$ with $5\le n \le 9$ . The coset intersection problem relative to $\Omega $ is to determine the family $\mathbf F$ of irreducible cubics in $PG(2,\mathbb K )$ for which $\Omega $ is a common coset of a subgroup of the additive group $(\mathcal F ,+)$ for every $\mathcal F \in \mathbf F$ . In this paper, a complete solution of this problem is given.     

Local maximal function and weights in a general setting

For a proper open set $\Omega $ immersed in a metric space with the weak homogeneity property, and given a measure $\mu $ doubling on a certain family of balls lying “well inside” of $\Omega $ , we introduce a local maximal function and characterize the weights $w$ for which it is bounded on $L^p(\Omega ,w d\mu )$ when $1<p<\infty $ and of weak type $(1,1)$ . We generalize previous known results and we also present an application to interior Sobolev’s type estimates for appropriate solutions of the differential equation $\Delta ^m u=f$ , satisfied in an open proper subset $\Omega $ of $\mathbb R ^n$ . Here, the data $f$ belongs to some weighted $L^p$ space that could allow functions to increase polynomially when approaching the boundary of $\Omega $ .     

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and $0\le \alpha \le 180^\circ $ , how large Hausdorff dimension can a compact set $A\subset \mathbb{R }^n$ have if $A$ does not contain three points that form an angle $\alpha $ ? (2) Given $\alpha $ and $\delta $ , how large Hausdorff dimension can a subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the $\delta $ -neighborhood of $\alpha $ ? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles $0$ and $180^\circ $ , the angles $60^\circ , 90^\circ $ and $120^\circ $ also play special role in problem (2): the maximal dimension is smaller for these angles than for the other angles. In problem (1) the angle $90^\circ $ seems to behave differently from other angles.     

Sharp energy estimates for nonlinear fractional diffusion equations

We study the nonlinear fractional equation $(-\Delta )^su=f(u)$ in $\mathbb R ^n,$ for all fractions $0<s<1$ and all nonlinearities $f$ . For every fractional power $s\in (0,1)$ , we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension $n=3$ whenever $1/2\le s<1$ . This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\mathbb R ^n$ . It remains open for $n=3$ and $s<1/2$ , and also for $n\ge 4$ and all $s$ .     

On the singular set of minimizers of p(x)-energies

We treat the partial regularity of locally bounded local minimizers $u$ for the $p(x)$ -energy functional $$\begin{aligned} \mathcal{E }(v;\Omega ) = \int \left( g^{\alpha \beta }(x)h_{ij}(v) D_\alpha v^i (x) D_\beta v^j (x) \right) ^{p(x)/2} dx, \end{aligned}$$ defined for maps $v : \Omega (\subset \mathbb R ^m) \rightarrow \mathbb R ^n$ . Assuming the Lipschitz continuity of the exponent $p(x) \ge 2$ , we prove that $u \in C^{1,\alpha }(\Omega _0)$ for some $\alpha \in (0,1)$ and an open set $\Omega _0 \subset \Omega $ with $\dim _\mathcal{H }(\Omega \setminus \Omega _0) \le m-[\gamma _1]-1$ , where $\dim _\mathcal{H }$ stands for the Hausdorff dimension, $[\gamma _1]$ the integral part of $\gamma _1$ , and $\gamma _1 = \inf p(x)$ .     

The equidistribution of small points for strongly regular pairs of polynomial maps

In this paper, we prove the equidistribution of periodic points of a regular polynomial automorphism $f : \mathbb{A }^n \rightarrow \mathbb{A }^n$ defined over a number field $K$ : let $f$ be a regular polynomial automorphism defined over a number field $K$ and let $v\in M_K$ . Then there exists an $f$ -invariant probability measure $\mu _{f,v}$ on $\mathrm{Berk }\bigl ( \mathbb{P }^n_\mathbb{C _v} \bigr )$ such that the set of periodic points of $f$ is equidistributed with respect to $\mu _{f,v}$ .     

2-Nilpotent real section conjecture

We show a $2$ -nilpotent section conjecture over $\mathbb{R }$ : for a geometrically connected curve $X$ over $\mathbb{R }$ such that each irreducible component of its normalization has $\mathbb{R }$ -points, $\pi _0(X(\mathbb{R }))$ is determined by the maximal $2$ -nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for $X$ smooth and proper, $X(\mathbb{R })^{\pm }$ is determined by the maximal $2$ -nilpotent quotient of $\mathrm{Gal }(\mathbb{C }(X))$ with its $\mathrm{Gal }(\mathbb{R })$ action, where $X(\mathbb{R })^{\pm }$ denotes the set of real points equipped with a real tangent direction, showing a $2$ -nilpotent birational real section conjecture.