Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $?h^2\mathrm{\Delta }+V(|x|)?E$ in dimension $n\ge 2$, where $h,E>0$, and $V:[0,\infty )\to \mathbb{R}$ is $L^{\infty }$ and compactly supported. The weighted resolvent norm grows no faster than $\exp ?(C{h}^{?1})$, while an exterior weighted norm grows $～h^{?1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.     

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type $y^2=x^3+f(t)x$, defined over the finite field ${\mathbb{F}}_q$ for a prime power $q\equiv 3\mathrm{mod}4$. It is shown that if $s^4=f(t)$ defines a curve that is maximal over ${\mathbb{F}}_{q^2}$ then the rank of the group of sections defined over ${\mathbb{F}}_q$ on the elliptic surface is determined in terms of elementary properties of the rational function $f(t)$. Similar results are shown for elliptic surfaces given by $y^2=x^3+g(t)$ using prime powers $q\equiv 5\mathrm{mod}6$ and curves $s^6=g(t)$. Finally, for each of the forms used here, existence of curves with the property that they are maximal over ${\mathbb{F}}_{q^2}$ is discussed, as well as various examples.     

On the structure of diffuse measures for parabolic capacities

Let $Q=(0,T)\times \mathrm{\Omega }$, where Ω is a bounded open subset of ${\mathbb{R}}^d$. We consider the parabolic p-capacity on Q naturally associated with the usual p-Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p-capacity, $p>1$, then it is of the form $\mu =f+\text{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in {(L^{p^{\prime }}(Q))}^d$ and $g\in L^p(0,T;W_0^{1,p}(\mathrm{\Omega })\cap L^2(\mathrm{\Omega }))$. We show the converse of this result: if $p>1$, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.     

On the boomerang uniformity of a class of permutation quadrinomials over finite fields

Let ${\mathbb{F}}_{2^n}$ be a finite field with $2^n$ elements and $f_{\underset{\_}{c}}(x)=c_0{x}^{2^m(2^k+1)}+c_1{x}^{2^{m+k}+1}+c_2{x}^{2^m+2^k}+c_3{x}^{2^k+1}\in {\mathbb{F}}_{2^n}[x]$, where n, m and k are positive integers with $n=2m$ and $\gcd ?(m,k)=e$. In this paper, motivated by a recent work of Li, Xiong and Zeng (Li et al. (2021) [12]), we further study the boomerang uniformity of $f_{\underset{\_}{c}}(x)$ by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result from the case of m being odd and $e=1$ to that of both $m/e$ and $k/e$ being odd.     

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations$\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u?\mathrm{\Delta }u+V(x)u?\frac{1}{2}u\mathrm{\Delta }(u^2)=g(u),\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2({\mathbb{R}}^N)\hfill \end{array}$ where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition $g(t)t\ge \mu G(t)\ge 0$ with $\mu >3$. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.