Non-null-controllability of the Grushin operator in 2D

We are interested in the exact null controllability of the equation ${?}_{t}f?{?}_x^{2}f?x^2{?}_y^{2}f={\mathbb{1}}_{\omega }u$, with control u supported on ω. We show that, when ω does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^2$ estimate on polynomials, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $?{?}_x^2+{(nx)}^2$ with Dirichlet conditions on $(?1,1)$, and we show a quite precise estimation it satisfies, even when n is in some complex domain.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

On a class of diagonal equations over finite fields

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\cdots +x_n^{2^m}=0$ over a finite field of characteristic $p\equiv \pm 3(\mathrm{mod}8)$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of p.     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

On the best constant in Gaffney inequality

We discuss the value of the best constant in Gaffney inequality namely

${6\mathrm{?}\omega 6}_{L^2}^2\le C({6d\omega 6}_{L^2}^2+{6\delta \omega 6}_{L^2}^2+{6\omega 6}_{L^2}^2)$

when either $\nu \wedge \omega =0$ or $\nu ?\omega =0$ on ?Ω.