ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p,q;Y}(\Omega ,ℬ)$..........................................................73. Some properties of the spaces $H_{p,q;Y}(\Omega ,ℬ)$...................................................104. Some examples of the spaces $H_{p,q;Y}(\Omega ,ℬ)$....................................................155....

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+u^{7/3}=0$ in $\Omega$, $u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain in ${ℝ}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_K>0$ such that for $0<\mu <{\mu }_K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

We consider digit expansions ${\sum }_{j=0}^{\ell -1}{\Phi }^j\left(d_j\right)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^1$-base and data in $h_c^1$, a subspace of $L$1. We derive our results, considering the action of an adjoint operator on $B_{T}MOC$, a predual of $h_c^1$, and using known properties of this last space.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺. ...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{|ℳfₙ}⇝T*T^{r*}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r*}M=J^r(M,ℝ)₀$ are classified. For n ≥ 3 all natural operators $T_{|ℳfₙ}⇝\Lambda ²T*T^{r*}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r*}M$ are completely described.

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in ${ℝ}^N$ with boundary of class C², $\alpha \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{\Omega }\alpha >0$, f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $su{p}_{\left|s\right|\le t}\left|g(\cdot ,s)\right|\in L^p\left(\Omega \right)$ and $g(\cdot ,t)\in L^{\infty }\left(\Omega \right)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1/2\xi ²-{\int }_0^{\xi }f\left(t\right)dt$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough,...

Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f\in C^{\infty }(M,ℝ)$ for which $P_{e^{f}g}$ has only simple non-zero eigenvalues is a residual set in $C^{\infty }(M,ℝ)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics...

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega$ whose boundary $\partial \Omega$ is formed by two circles ${\Gamma }_1$, ${\Gamma }_2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho$, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$). ...