W01,1 solutions in some borderline cases of Calderon–Zygmund theory

In this paper we study the existence of $W_0^{1,1}(\Omega )$ distributional solutions of Dirichlet problems whose simplest example is$\{\begin{array}{cc}?\mathrm{div}\left({|\mathrm{?}u|}^{p?2}\mathrm{?}u\right)=f(x),\hfill & \text{in}\Omega ;\hfill \\ u=0,\hfill & \text{on}?\Omega .\hfill \end{array}$     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes

In the present paper we perform the homogenization of the semilinear elliptic problem

$\{\begin{array}{cc}{u}^{\epsilon }\ge 0\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ ?divA(x)D{u}^{\epsilon }=F(x,u^{\epsilon })\hfill & \text{in}{\mathrm{\Omega }}^{\epsilon },\hfill \\ u^{\epsilon }=0\hfill & \text{on}?{\mathrm{\Omega }}^{\epsilon }.\hfill \end{array}$

In this problem $F(x,s)$ is a Carathéodory function such that $0\le F(x,s)\le h(x)/\mathrm{\Gamma }(s)$ a.e. $x\in \mathrm{\Omega }$ for every $s>0$, with h in some $L^r(\mathrm{\Omega })$ and Γ a $C^1([0,+\infty [)$ function such that $\mathrm{\Gamma }(0)=0$ and ${\mathrm{\Gamma }}^{\prime }(s)>0$ for every $s>0$. On the other hand the open sets ${\mathrm{\Omega }}^{\epsilon }$ are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function $F(x,s)$ satisfies $0\le F(x,s)\le h(x)(\frac{1}{s}+1)$. In this case the solution $u^{\epsilon }$ to the problem belongs to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ and its definition is a “natural” and rather usual one.In the general case where $F(x,s)$ exhibits a “strong singularity” at $u=0$, which is the purpose of the present paper, the solution $u^{\epsilon }$ to the problem only belongs to $H_{\text{loc}}^1({\mathrm{\Omega }}^{\epsilon })$ but in general does not belong to $H_0^1({\mathrm{\Omega }}^{\epsilon })$ anymore, even if $u^{\epsilon }$ vanishes on $?{\mathrm{\Omega }}^{\epsilon }$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” $\mu u^0$ still appears in the left-hand side while the source term $F(x,u^0)$ is not modified in the right-hand side.     

Topological bounds for Fourier coefficients and applications to torsion

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded convex domain in the plane and consider

$\begin{array}{cc}\hfill ?\mathrm{\Delta }u& =1\text{in}\mathrm{\Omega }\hfill \\ \hfill u& =0\text{on}?\mathrm{\Omega }.\hfill \end{array}$

If u assumes its maximum in $x_0\in \mathrm{\Omega }$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^{2}u(x_0)$. We prove that $D^{2}u(x_0)$ is negative definite and give a quantitative bound on the spectral gap

$\begin{array}{c}{\lambda }_{\max }(D^{2}u(x_0))\le ?c_1\exp ?(?c_2\frac{\mathrm{diam}(\mathrm{\Omega })}{\mathrm{inrad}(\mathrm{\Omega })})\hfill \\ \text{for universal}{c}_1,c_2>0.\hfill \end{array}$

This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T}\to \mathbb{R}$ is continuous and has n sign changes, then

$\sum _{k=0}^{n/2}\left|〈f,\sin ?kx〉\right|+\left|〈f,\cos ?kx〉\right|{?}_n\frac{{|f6}_{L^1(\mathbb{T})}^{n+1}}{{6f6}_{L^{\infty }(\mathbb{T})}^n}.$

This statement immediately implies estimates on higher derivatives of harmonic functions u in the unit ball: if u is very flat in the origin, then the boundary function $u(\cos ?t,\sin ?t):\mathbb{T}\to \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T}\to \mathbb{R}$ cannot decay faster than $～\exp ?(?{(\mathrm{\#}\text{sign changes})}^{2}t/4)$.     

Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient $f(z)$ of the critical nonlinearity. For sufficiently small $\lambda ,\mu >0$, there are at least k positive solutions of the semilinear elliptic systems$\{\begin{array}{c}?\mathrm{\Delta }u=\lambda g(z){|u|}^{p?2}u+\frac{\alpha }{\alpha +\beta }f(z){|u|}^{\alpha ?2}u{|v|}^{\beta }\text{in}\Omega ;\hfill \\ ?\mathrm{\Delta }v=\mu h(z){|v|}^{p?2}v+\frac{\beta }{\alpha +\beta }f(z){|u|}^{\alpha }{|v|}^{\beta ?2}v\text{in}\Omega ;\hfill \\ u=v=0\text{on}?\Omega ,\hfill \end{array}$ where $0\in \Omega ?{\mathbb{R}}^N$ is a bounded domain, $\alpha >1$, $\beta >1$ and $2<p<\alpha +\beta =2^{?}$ for $N>4$.     

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $PSL(2,\mathbb{Z})?\mathbb{H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma ?\mathbb{H}$ and odd and grotesque continued fractions, where $\Gamma ?{\mathbb{Z}}_{3}1{\mathbb{Z}}_3$ is the index two subgroup of $PSL(2,\mathbb{Z})$ generated by the order three elements $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill ?1\hfill & \hfill 1\hfill \end{array}\right)$, and having an ideal quadrilateral as fundamental domain.A similar connection between geodesics on $\Theta ?\mathbb{H}$ and even continued fractions is discussed in our framework, where $\Theta$ denotes the Theta subgroup of $PSL(2,\mathbb{Z})$ generated by $\left(\begin{array}{cc}\hfill 0\hfill & \hfill ?1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ and $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.