In this paper we study local regularity for the generalized solution to the Dirichlet problem related to the equation $$Lu\equiv X_i^{*}(a_{ij}{X}_{j}u)=f.$$ where $X_1,X_2,\mathrm{\dots },X_m$ are vector fields satisfying Hörmander condition and $a_{ij}\in L^{\mathrm{\infty }}$. We give a representation formula for the generalized solution in terms of the Green function and thanks to suitable estimates we achieve our goal. In the case $f\ge 0$ we are able to give necessary condition too.

L’equazione $u_{tt}={\sum }_{ij=1}^n{(a_{ij}(x,t)u_{x_j})}_{x_i}$ in condizioni di debole iperbolicità $\left({\sum }_{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\ge 0\right)$, è ben posta negli spazi di Gevrey ${\gamma }_{loc}^{(s)}$ con , purché $a_{ij}$ sia di Gevrey in $x$ di ordine $s$ e risulti $${\left[\sum _{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\right]}^{1/\sigma }\in BV([0,T]:{𝐋}_{loc}^{\mathrm{\infty }})$$

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 $m\ge 11$.

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...

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$ $[\sigma \left(n\right)={min}_{1\le i\le m}{\sigma }_i\left(n\right)]$. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

Let $\mathrm{\Omega }$ be a bounded open subset of $R^n$, $n>4$, of class $C^2$ . Let $u\in H^2\left(\mathrm{\Omega }\right)$ a solution of elliptic non linear non variational system $$a\left(x,u,Du,H\left(u\right)\right)=b\left(x,u,Du\right)$$ where $a\left(x,u,\mu ,\xi \right)$ and $b\left(x,u,\mu \right)$ are vectors in $R^N$, $N\ge 1$, measurable in $x$, continuous in $\left(u,\mu ,\xi \right)$ and $\left(u,\mu \right)$ respectively. Here, we demonstrate that if $b\left(x,u,\mu \right)$ has limit controlled growth, if $a\left(x,u,\mu ,\xi \right)$ is of class $C^1$ in $\xi$ and satisfies the Campanato condition $\left(A\right)$ and, together with $\frac{\partial a}{\partial \xi }$, certain continuity assumptions, then the vector $Du$ is partially Hölder continuous for every exponent $\alpha <1-\frac{n}{p}$.

We provide a duality theory and existence results for a operator equation $\nabla T(x)=\nabla N(x)$ where $T$ is not necessarily a monotone operator. We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.

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 $s$.

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. ...

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 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 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 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...

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove inverse approximation theorems for functions $f$ such that $w_{2m}f\in L^p(R)$, where $1\le p\le \mathrm{\infty }$ and $w_{2m}(x)={(1+x^{2m})}^{-1}$, $m\in N_0$.

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.

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...

We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as $$-\mathrm{div}a(x,\nabla u)=b(x,\nabla u)+\mu$$ where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$, $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^1$ function, we prove a uniqueness result under more restrictive hypotheses on the operator.

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...

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 $X\subset {\mathbb{P}}^N$ 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...

We use notations: $[x,y]=[x_{,1}y]$ and $[x_{,k+1}y]$ e $[[x_{,k}y],y]$. We consider groups generated by $x$, $y$ satisfying relations $x=[x_{,n}y],y=[y_{,n}x]$ or $[x,y]=[x_{,n}y]$, $[y,x]=[y_{,n}x]$. We call groups of the first type mep-groups and of the second type nmep-groups. We show many properties and examples of mep- and nmep-groups. We prove that if $p$ is a prime then the group $S{l}_2(p)$ is a nmep-group. We give the necessary and sufficient conditions for metacyclic group to be a nmep-group and we show that nmep-groups with presentation $⟨x,y\mid [x,y]=[x_{,2}y],[y,x]=[y_{,2}x],x^n,y^m⟩$ are finite.

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.

L’equazione $u_{tt}={\sum }_{ij=1}^n{(a_{ij}(x,t)u_{x_j})}_{x_i}$ in condizioni di debole iperbolicità $\left({\sum }_{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\ge 0\right)$, è ben posta negli spazi di Gevrey ${\gamma }_{loc}^{(s)}$ con , purché $a_{ij}$ sia di Gevrey in $x$ di ordine $s$ e risulti $$\left[\sum _{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\right]{}^{1/\sigma }\in BV([0,T]:{𝐋}_{loc}^{\mathrm{\infty }})$$

We introduce a new class of spaces, called Hausdorff modulo $ℐ$ or $T_2$ mod $ℐ$ spaces with respect to an ideal $ℐ$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $ℐ$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $ℐ$-compact subsets in Hausdorff modulo $ℐ$ spaces and generalize some results of Hamlett and Jankovic....

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.

The aims of the present paper can be described as follows: a) In [2] Beauville showed that if some endomorphism $u$ a Jacobian $J(C)$ has connected kernel, the principal polarization on $J(C)$ induces a multiple of the principal polarization on the image of $u$. We reformulate and complete this theorem proving "constructively" the following: Theorem. Let $Z\subset J(C)$ be an abelian subvariety and $Y$ its complementary variety. $Z$ is a Prym-Tyurin variety with respect to $J(C)$ if and only if the following sequence...

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.

If the second order problem $\ddot{u}+\dot{u}+Au=f$ has $L^p$ maximal regularity for some $p\in (1,\mathrm{\infty })$, then it has $L^p$ maximal regularity for every $p\in (1,\mathrm{\infty })$.

In this note we show a relative version of ${\gamma }_k$-set introduced and studied in [12]. We give several characterizations of this property; in particular one of the characterizations is Ramsey theoretical. Also we give a result involving a property of the corresponding mapping between function spaces.

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...

Let $f(\cdot ,t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$, with initial data $f_0$, and let $g_{\sigma }$ be the Gaussian density with zero mean and variance ${\sigma }^2$, ${\sigma }^2$ being the value of the second moment of $f_0$. Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot ,t)$ and $g_{\sigma }$ goes to zero, as $t\to +\mathrm{\infty }$, with an exponential rate equal to $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture,...

The range of the bigenus $P_2$ is one of the unsolved problems concerning smooth complex projective regular threefolds of general type with $p_g=0$: The examples in the literature have $P_2\le 5$. In the present paper we present a non-singular threefold with $p_g=q_1=q_2=0$; $P_2=6$; the bicanonical map is stably birational.

Let $E$ be a finitely generated $R$-module, having finite projective dimension. We study the acyclicity of the approximation complex $𝒵(E)$ of $E$ in terms of certain Fitting conditions $F_k^{(i)}$ on the Fitting ideals of the $i$-th module of a projective resolution of $E$. We deduce some good properties of the symmetric algebra of $E$.

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...