A singular Liouville equation on planar domains

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, $-\mathrm{\Delta }u=(\log u+f(u)){\chi }_{\{u>0\}}$ in $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional $I_{\epsilon }$ corresponding to the perturbed equation $-\mathrm{\Delta }u+g_{\epsilon }(u)=f(u)$, where $g_{\epsilon }$ is well defined at 0 and approximates $-\log u$. We show that $I_{\epsilon }$ has a critical point $u_{\epsilon }$ in $H_0^1(\mathrm{\Omega })$, which converges to a legitimate nontrivial nonnegative solution of the original problem as $\epsilon \to 0$. We also investigate the problem with $f(u)$ replaced by $\lambda f(u)$, when the parameter $\lambda >0$ is sufficiently large.     

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

We prove $L^p$ estimates, $1\le p\le \infty$, for solutions to the tangential Cauchy–Riemann equations ${\overline{?}}_{b}u=?$ on a class of infinite type domains $\mathrm{\Omega }?{\mathbb{C}}^2$. The domains under consideration are a class of convex ellipsoids, and we show that if $?$ is a ${\overline{?}}_b$‐closed (0,1)‐form with coefficients in $L^p$, then there exists an explicit solution u satisfying ${\parallel u\parallel }_{L^p(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^p(b\mathrm{\Omega })}$. Moreover, when $p=\infty$, we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the $\overline{?}$‐equation, that is, given a smooth (0,1)‐form ? on $b\mathrm{\Omega }$ with an L¹‐boundary value, we can solve the Cauchy–Riemann equation $\overline{?}u=?$ so that ${\parallel u\parallel }_{L^1(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^1(b\mathrm{\Omega })}$ where C is independent of $u$ and ?. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.     

Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Single-peak solutions for a subcritical Schrödinger equation with non-power nonlinearity

This paper deals with the following slightly subcritical Schrödinger equation: $$-\mathrm{\Delta }u+V(x)u=f_{\epsilon }(u),u>0\text{in}{\mathbb{R}}^N,$$ where $V(x)$ is a nonnegative smooth function, $f_{\epsilon }(u)=\frac{u^p}{{[\ln (e+u)]}^{\epsilon }}$, $p=\frac{N+2}{N-2}$, $\epsilon >0$, $N\ge 7$. Most of the previous works for the Schrödinger equations were mainly investigated for power-type nonlinearity. In this paper, we will study the case when the nonlinearity $f_{\epsilon }(u)$ is a non-power nonlinearity. We show that, for ε small enough, there exists a family of single-peak solutions concentrating at the positive stable critical point of the potential $V(x)$.     

Noncompact quasi‐Einstein manifolds conformal to a Euclidean space

The goal of this article is to investigate nontrivial m‐quasi‐Einstein manifolds globally conformal to an n‐dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n?1)$‐dimensional translation group, we provide a complete classification when $\lambda =0$ and $m\ge 1$ or $m=2?n$.     

The fractional porous medium equation on manifolds with conical singularities II

This is the second of a series of two papers that studies the fractional porous medium equation, ${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma }{(|u|}^{m-1}u)=0$ with $m>0$ and $\sigma \in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_{\infty }-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.     

Gauss-Prym maps on Enriques surfaces

We prove that the kth Gaussian map ${\gamma }_H^k$ is surjective on a polarized unnodal Enriques surface $(S,H)$ with $\phi (H)>2k+4$. In particular, as a consequence, when $\phi (H)>4(k+2)$, we obtain the surjectivity of the kth Gauss-Prym map ${\gamma }_{{\omega }_C\otimes \alpha }^k$, with $\alpha :={\omega }_{{S|}_C}$, on smooth hyperplane sections $C\in |H|$. In case $k=1$, it is sufficient to ask $\phi (H)>6$.     

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

For a positive integer N, let $X_0(N)$ be the modular curve over $\mathbf{Q}$ and $J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_0(N)$ is equal to the rational cuspidal divisor class group of $X_0(N)$ when $N=p^{2}M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.     

Generalization of Hardy–Littlewood maximal inequality with variable exponent

Let $p(·)$ be a measurable function defined on ${\mathbb{R}}^d$ and $p_{-}:={inf}_{x\in {\mathbb{R}}^d}p(x)$. In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone-like set defined by a given function ψ. Moreover, instead of the integral means, we consider the $L_{q(·)}$-means. Let $p(·)$ and $q(·)$ satisfy the log-Hülder condition and $p(·)=q(·)r(·)$. Then, we prove that the maximal operator is bounded on $L_{p(·)}$ if and is bounded from $L_{p(·)}$ to the weak $L_{p(·)}$ if $1\le r_{-}\le \infty$. We generalize also the theorem about the Lebesgue points.     

Improved Bohr inequality for harmonic mappings

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk $\mathbb{D}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part h is bounded by 1 and that $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in $\mathbb{D}$ and for some $k\in [0,1]$.     

Resolvent family for the evolution process with memory

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, $\{(G(t),F(t)),t\ge 0\}$, that is, called the resolvent family for the linear evolution process with memory, the $F(t)$ is called the memory effect family. In this paper, we prove that the families $G(t)$ and $F(t)$ are exponentially bounded, and the family $(G(t),F(t))$ associate with an operator pair $(A,L)$ that is called generator of the resolvent family. Using $(A,L)$, we derive associated differential equation with memory and representation of $F(t)$ via L. These results give necessary conditions of the well-posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on $(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro-differential equation and functional differential equations.     

Invariant hypercomplex structures and algebraic curves

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{g}\mathfrak{l}(k,\mathbb{C})$ correspond to algebraic curves C of genus ${(k-1)}^2$, equipped with a flat projection $\pi :C\to {\mathbb{P}}^1$ of degree k, and an antiholomorphic involution $\sigma :C\to C$ covering the antipodal map on ${\mathbb{P}}^1$.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.