In this paper we prove the existence of a solution for a problem whose model is: with $f(x)$ in $L^1(\mathrm{\Omega })$ and $\sigma$, $\gamma >0$.

Assume that $\mathrm{\Omega }$, ${\mathrm{\Omega }}^{\prime }$ are planar domains and $f:\mathrm{\Omega }\stackrel{\text{onto}}{\to }{\mathrm{\Omega }}^{\prime }$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\mathrm{\Omega };{ℝ}^2)$ with finite distortion. We prove that if the distortion function $K_f$ of $f$ satisfies the condition , then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L_{\text{loc}}^1({\mathrm{\Omega }}^{\prime })$. We show that this result is sharp in sense that the conclusion fails if ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=1$. Moreover, we prove that if the distortion function $K_f$ satisfies the condition ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=\lambda$ for some $\lambda >0$, then $K_{f^{-1}}$ belongs to $L_{\text{loc}}^p({\mathrm{\Omega }}^{\prime })$ for every $p\in (0,\frac{1}{2\lambda })$. As special case of this result we show that if...

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution $u$ in a singular way and on its gradient $Du$ with quadratic growth. The prototype of the problem under consideration is where $\lambda >0$, $k>0$; $f(x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, $f(x)\ge 0$ (and so $u\ge 0$). If , we prove the existence of a solution for both the "+" and the "-" signs, while if $k\ge 1$, we prove the existence of a solution for the "+" sign only.

In this paper we study the nonlinear Schrödinger-Maxwell equations If $V$ is a positive constant, we prove the existence of a ground state solution $(u,\varphi )$ for . The non-constant potential case is treated for , and $V$ possibly unbounded below.

Let us consider a Young's function $\mathrm{\Phi }:{ℝ}^{+}\to {ℝ}^{+}$ satisfying the ${\mathrm{\Delta }}_2$ condition together with its complementary function $\mathrm{\Psi }$, and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: $B$ the unit ball of ${ℝ}^n$. In this paper we give a necessary and sufficient condition for the $L^{\varphi }$-solvability of the problem, where $L^{\varphi }$ is the Orlicz Space generated by the function $\mathrm{\Phi }$. This means solvability for $f\in L^{\mathrm{\Phi }}$ in the sense of [5], [8], where the case $\mathrm{\Phi }(t)=t^p$ is treated. ...

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\>0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \>0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

The Tango bundle $T$ is defined as the pull-back of the Cayley bundle over a smooth quadric $Q_5$ in ${\mathbb{P}}_6$ via a map $f$ existing only in characteristic 2 and factorizing the Frobenius $\phi$. The cohomology of $T$ is computed in terms of $S\otimes C$, ${\phi }^{*}(C)$, ${\text{Sym}}^2(C)$ and $C$, which we handle with Borel-Bott-Weil theorem.

Let $I=[a,b]\subset ℝ$, let , let $u$ and $v$ be positive functions with $u\in L_{p^{\prime }}(I)$ e $v\in L_q(I)$ and let $T:L_p(I)\to L_q(I)$be the Hardy-type operator given by $$\left(Tf\right)\left(x\right)=v\left(x\right){\int }_a^{x}f\left(t\right)u\left(t\right)dt,x\in I.$$ We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system $$g\left(x\right)=\left(TF\right)\left(x\right){\left(f\left(x\right)\right)}_{\left(p\right)}=\lambda \left(T^{*}{g}_{\left(p\right)}\right))\left(x\right)$$ (where, for example,$t_{(p)}={|t|}^{p-1}\mathrm{sgn}(t)$is given by $$\begin{array}{cc}& \underset{n\to \infty }{lim}n{\widehat{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^r{)}^{1/r}dt\right)}^{1/r},\text{for}1<p<q<\infty \hfill \\ & \underset{n\to \infty }{lim}n{\check{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^{r}dt\right)}^{1/r}\text{for}1<q<p<\infty \hfill \end{array}$$ Here$r=\frac{1}{p^{\prime }}+\frac{1}{p}$, $c_{p,q}$ is an explicit constant depending only on $p$ and $q$, $\widehat{\lambda }(T)=\max (s{p}_n(T,p,q))$, ${\check{\lambda }}_n(T)=\min (s{p}_n(T,p,q))$ where $s{p}_n(T,p,q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.

We investigate the following quasilinear and singular problem, $$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\&gt;0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$ where $\Omega$ is an open bounded domain with smooth boundary, $1\&lt;p\&lt;\infty$, $p-1\&lt;q\le p^{*}-1$, $\lambda \&gt;0$, and $0\&lt;\delta \&lt;1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\&lt;p\&lt;N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\&gt;N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Let ${𝕊}^1$ and $𝔻$ be the unit circle and the unit disc in the plane and let us denote by $𝒜({𝕊}^1)$ the algebra of the complex-valued continuous functions on ${𝕊}^1$ which are traces of functions in the Sobolev class $W^{1,2}(𝔻)$. On $𝒜({𝕊}^1)$ we define the following norm $$\parallel u\parallel ={\parallel u\parallel }_{L^{\mathrm{\infty }}({𝕊}^1)}+{\left({\iint }_{𝔻}{|\nabla \tilde{u}|}^2\right)}^{1/2}$$ where is the harmonic extension of $u$ to $𝔻$. We prove that every isomorphism of the functional algebra $𝒜({𝕊}^1)$ is a quasitsymmetric change of variables on ${𝕊}^1$.

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following...

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that Im $z$ Im $f\left(z\right)$ $\ge 0$ for $z\in \Delta$. Moreover, let us denote: ${\mathrm{T}}^{\left(2\right)}:=\{f\in \mathrm{T}:f\left(z\right)=-f(-z)\text{for}z\in \Delta \}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}$, which we denote...

The object of this note is to study certain 2-dimensional $\lambda$-adic representations of $Gal(\overline{Q}/Q)$; fixed $p_1,\mathrm{\dots },p_n$ distinct primes, we will consider representations $\rho :G\to G{L}_2(A)$, given by the matrix which are unramified outside $p_1,\mathrm{\dots },p_n,\mathrm{\infty }$ and the residue characteristic of $\lambda$, which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$. In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke...

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus \{0\}$, $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every...

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

We attempt to generalize conductor degree's results, known in ${\mathbb{P}}^2$, to the case of 0-dimensional schemes of ${\mathbb{P}}^r$. In the first part of this paper, we consider the problem of characterizing the sequences generators's degrees of the conductor which are compatible with a fixed postulation (or Hilbert function) for a set of points in ${\mathbb{P}}^r$ and we determine the conductor degree of every point in a $r$-partial intersection. In addition, we define the separating degree of a point for a 0-dimensional subscheme...

The aim of this work is to prove a chain rule and an $L^1$-lower semicontinuity theorems for integral functional defined on $BV(\mathrm{\Omega })$. Moreover we apply this result in order to obtain new relaxation and $\mathrm{\Gamma }$-convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$.

The rational completion $\overline{M}$ of an $R$-module $M$ can be characterized as a ${\tau }_M$-injective hull of $M$ with respect to a (hereditary) torsion functor ${\tau }_M$ depending on $M$. Properties of a torsion functor depending on an $R$-module $M$ are studied.

This article is devoted to sequences ${(u_n)}_n$ of subharmonic functions in ${ℝ}^N$, with finite order, whose means $J_{u_n}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r>0$, $\exists A_r>0$ such that $J_{u_n}(r)\le A_r$, $\forall n\in 𝐍$. The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.

We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }|̸M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type at $\mathrm{\ell }$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global...

Using the theory of adjoints and pluricanonical adjoints, we construct three nonsingular threefolds, as desingularizations of degree six hypersurfaces in ${\mathbb{P}}^4$, having the irregularities $q_1=q_2=0$ and the following periodical sequences of plurigenera respectively $$(p_g,P_2,P_3,\mathrm{\dots },P_m,\mathrm{\dots })=(0,0,1,0,0,1,\mathrm{\dots }),(0,0,0,1,0,0,0,1,\mathrm{\dots }),(0,0,0,0,1,0,0,0,0,1,\mathrm{\dots }).$$In the Appendix, starting from the second above-mentioned example, we construct a threefold of general type with $q{q}_1=q_2=0,p_g=1$, $P_2=2$ whose m-canonical transformation is birational if and only if $m\ge 11$.

If $S$ is a pseudo-symmetric numerical semigroup, $g$ is its Frobenius number and $S$ is a minimal generator of $S$, then $S\cup \{g\}$, $S\setminus \{g\}$S and $S\cup \{\frac{1}{2}g,g\}$ are also numerical semigroups. In this paper we study these constructions.

We consider an SPDE in a Hilbert space $H$ of the form $dX(t)=(AX(t)+b(X(t)))dt+\sigma (X(t))dW(t)$, $X(0)=x\in H$ and the corresponding transition semigroup $P_{t}f(x)=𝔼[f(X(t,x))]$. We define the infinitesimal generator $\overline{L}$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\phi =\frac{1}{2}\mathrm{Tr}[\sigma (x){\sigma }^{*}(x)D^2\phi ]+⟨b(x),D\phi ⟩$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\overline{L}$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular...

We study the irreducibility of the Dirichlet polynomial $P_G(s)$ when $G$ is the alternating group on $p+1$ elements with $p$ prime and we prove that $P_G(s)$ is irreducible for infinitely many choiches of $p$.

Let $X\subset {\mathbb{P}}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N\ge 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X)=g(X)-1$ are the del Pezzo surface in ${\mathbb{P}}^4$, in ${\mathbb{P}}^5$ and a conic bundle of degree 5 in ${\mathbb{P}}^4$. We prove that for $N\ge 5$ if $PG(X)=g(X)-1+\lambda$, $\lambda$ a non negative integer, then $g(X)\le \lambda +1+\alpha$ where...

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}\left(V\right):=dim{H}^0(V,12K_V)\&gt;0$ and $P_{m_0}\left(V\right)\&gt;1$ for some positive integer $m_0\le 24$. A direct consequence is the birationality of the pluricanonical map ${\varphi }_m$ for all $m\ge 126$. Besides, the canonical volume $\text{Vol}\left(V\right)$ has a universal lower bound $\nu \left(3\right)\ge \frac{1}{63·{126}^2}$.

Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality: $$\text{Br}\left(X\right)\times \text{Pic}\left(X\right)\to \mathbb{Q}/\mathbb{Z}$$ between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\text{Br}\left(X\right)$ in ${\displaystyle \prod _{P\in X}\text{Br}\left(k_P\right)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\tilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

We prove that the $\mathrm{\Gamma }$-limit in $L_{\mu }^{\mathrm{\infty }}$ of a sequence of supremal functionals of the form $F_k(u)={\mu -\mathrm{ess}\sup }_{\mathrm{\Omega }}{f}_k(x,u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\mathrm{\Gamma }$-lim $F(\cdot ,B)$ of a sequence of functionals $F_k(u,B)={\mu -\mathrm{ess}\sup }_B{f}_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u,B)={\mu -\mathrm{ess}\sup }_{B}f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope. ...

We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $\ell$ of terms. Then we look at the possible decompositions $f\left(x\right)=g\left(h\right(x\left)\right)$, where $g,h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $\ell$ (and we provide explicit bounds). This supports and quantifies...

We consider the Grassmannian $𝔾r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\mathbb{P}}^1,{𝒪}_{{\mathbb{P}}_1}(n))$. We define ${𝔛}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on ${\mathbb{P}}^1$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of ${𝔛}_{k,r,d}$. In the second part we make a link between ${𝔛}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...