In this note we obtain the existence and the uniqueness of the solution of a variational inequality associated to the degenerate operator $$Lu=-\sum _{i,j=1}^n{(a_{ij}(x)u_{x_i}+d_{j}u)}_{x_j}+\sum _{i=1}^n{b}_i{u}_{x_i}+cu$$ assuming the coefficients of the lower terms and the known term belonging to a suitable degenerate Stummel-Kato class. The weight $w$, which gives the degeneration, belongs to the Muckenoupt class $A^2$.

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.

In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $div(A(x)\nabla u(x))=0$, where the symmetric matrix $A(x)$ satisfies the condition ${|\xi |}^2+{|A(x)\xi |}^2\le K(x)⟨A(x)\xi ,\xi ⟩$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp (\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L_{\text{loc}}^2{\log }^{\alpha -1}L$, . Moreover we show exactly how the degree of improved regularity depends on $\beta$.

In the present work we prove the interior Hölder continuity of the gradient matrix of any weak solution of equations, which describes the motion of non-Newtonian fluid in two dimensions, restricting ourself to the shear thinning case .

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 characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

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 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.

We prove a result about the rectifiability of class $C^2$ of the set of regular values (in the sense of Clarke) of a Lipschitz map $\phi :{ℝ}^n\to {ℝ}^N$ with

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...

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 global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $divA(x,u,Du)+B(x,u,DU)=0$, under natural polynomial growth of the coefficient functions $A$ and $B$. We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

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 $D$ 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...

Let $\gamma ,\tau :[a,b]\to R^{k+1}$ be a couple of Lipschitz maps such that ${\gamma }^{\prime }=\pm |{\gamma }^{\prime }|\tau$ almost everywhere in $[a,b]$. Then $\gamma ([a,b])$ is a $C^2$-rectifiable set, namely it may be covered by countably many curves of class $C^2$ embedded in $R^{k+1}$. As a conseguence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a $C^2$-rectifiable set.

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 $p\in (1,\mathrm{\infty })$.

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.

Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell’equazione di evoluzione non autonoma $u^{\prime }(t)\mathrm{—}A(t)u(t)=f(t)$, con il dato iniziale $u(0)=x$, in spazi di Banach. I dominii degli operatori $A(t)$ variano in $t$ e non sono necessariamente densi in $E$. Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità holderiana della soluzione e della sua derivata.

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.

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. ...

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.

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 ${𝕊}^1$.

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 $p$.

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 $$\left[\sum _{ij=1}^n{a}_{ij}(x,t){\xi }_i{\xi }_j\right]{}^{1/\sigma }\in BV([0,T]:{𝐋}_{loc}^{\mathrm{\infty }})$$

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 $\sigma$, $\gamma >0$.

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 $w_{2m}(x)={(1+x^{2m})}^{-1}$, $m\in N_0$.

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 $X\subset {\mathbb{P}}^N$ 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 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 give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}$ whenever the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot )$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_a$, , maps $L^{p(\cdot )}$ into $L^{q(\cdot )}$, where $1/p(\cdot )-1/q(\cdot )=a/n$.

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....

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.

Se il problema di minimo $({𝒫}_{\mathrm{\infty }})$ è il limite, in senso variazionale, di una successione di problemi di minimo con ostacoli del tipo $$\underset{u\ge {\psi }_h}{\min }{\int }_A\left[f_h(x,u,Du)+b(x,u)\right]𝑑x,$$ allora $({𝒫}_{\mathrm{\infty }})$ può essere scritto nella forma $$\underset{u}{\min }\left\{{\int }_A\left[f_{\mathrm{\infty }}(x,u,Du)+b(x,u)\right]𝑑x+{\int }_A{g}_{\mathrm{\infty }}(x,\tilde{u}(x))𝑑{\lambda }_{\mathrm{\infty }}(x)\right\}$$ dove $\tilde{u}$ è un conveniente rappresentante di $u$ e ${\lambda }_{\mathrm{\infty }}$ è una misura non negativa.

Diamo condizioni sulle funzioni $f$, $g$ e sulla misura $\mu$ affinché il funzionale $$F(u)={\int }_{\mathrm{\Omega }}f(x,u,Du)𝑑x+{\int }_{\overline{\mathrm{\Omega }}}g(x,u)𝑑\mu$$ sia $L^1(\mathrm{\Omega })$-semicontinuo inferiormente su $W^{1,1}(\mathrm{\Omega })\cap C^0(\overline{\mathrm{\Omega }})$. Affrontiamo successivamente il problema del rilassamento.

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 $I=[a,b]\subset ℝ$, 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 prove the existence of rotating solitary waves (vortices) for the nonlinear Klein-Gordon equation with nonnegative potential, by finding nonnegative cylindrical solutions to the standing equation where $x=(y,z)\in {ℝ}^k\times {ℝ}^{N-k}$, $N>k\ge 2$, $\mu >0$ and $\lambda \ge 0$. The nonnegativity of the potential makes the equation suitable for physical models and guarantees the wellposedness of the corresponding Cauchy problem, but it prevents the use of standard arguments in providing the functional associated to $(†)$ with bounded Palais-Smale...

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_k\left(X\right)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_k\left(X\right)$ is Ascoli iff $C_k\left(X\right)$ is a $k_{ℝ}$-space iff X is locally compact. Moreover, $C_k\left(X\right)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

In this paper we study the evolutive problem of linear viscoelasticity with a singular kernel memory $G^{\prime }$. We assume that $G^{\prime }$ presents an initial singularity, so that it is not a $L^1$-function in time, whereas the relaxation function $G$ is integrable at $t=0$. By applying the Fourier transform method, we prove a theorem of existence and uniqueness of the weak solutions in a functional space whose definition is strictly related to the properties of the kernel memory.

Sfruttando i risultati di [1], si prova che le derivate spaziali $D^{\alpha }u$ di ordine $|\alpha |$ con delle soluzioni in $Q$ di un sistema parabolico quasilineare di ordine $2m$ con andamenti strettamente controllati, sono parzialmente hölderiane in $Q$ con esponente di hölderianità decrescente al crescere di $|\alpha |$.

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...