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....

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work,...

The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of...

In this paper, we analyze and characterize all solutions about $\alpha$-migrativity properties of the five subclasses of 2-uninorms, i. e. $C^k$, $C_k^0$, $C_k^1$, $C_1^0$, $C_0^1$, over semi-t-operators. We give the sufficient and necessary conditions that make these $\alpha$-migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for $G\in C^k$, the $\alpha$-migrativity of $G$ over a semi-t-operator $F_{\mu ,\nu }$ is closely related to the $\alpha$-section of $F_{\mu ,\nu }$ or the ordinal sum representation...

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.

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $\mathscr{H}$ be a Hilbert space and let $A$ be a positive bounded operator on $\mathscr{H}$. The semi-inner product ${\langle h\mid k\rangle }_A:=\langle Ah\mid k\rangle$, $h,k\in \mathscr{H}$, induces a semi-norm ${\parallel ·\parallel }_A$. This makes $\mathscr{H}$ into a semi-Hilbertian space. An operator $T\in {ℬ}_A\left(\mathscr{H}\right)$ is said to be $(n,m)$-$A$-normal if $[T^n,{\left(T^{{♯}_A}\right)}^m]:=T^n{\left(T^{{♯}_A}\right)}^m-{\left(T^{{♯}_A}\right)}^m{T}^n=0$ for some positive integers $n$ and $m$.

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.

Let $D_m$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^{2}x+m^2$, where $m$ is an integer. In this paper we prove that the two points $P_{-1}=(-m,m)$ and $P_0=(0,m)$ on $D_m$ can be extended to a basis for $D_m\left(\mathbb{Q}\right)$ under certain conditions described explicitly.

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.

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

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...

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...