On the Existence and Stability of Solutions for a System of Elliptic Equations

We show, using a dual variational method developed in the paper, the existence of solutions for a system of Dirichlet problems for partial differential equations involving p– and q–Laplacian. The stability of solutions and the existence of positive solutions is considered in the superlinear case.      

(n+1)重周期和准周期Riemann边值问题

该文首先考虑了Rⁿ⁺¹维欧氏空间中(n+1)重周期正则函数和(n+1)重准周期正则函数的一些性质, 然后分别讨论了n+1重周期和准周期Riemann边值问题, 分别给出了两种边值问题 解的显式表达式和可解条件.     

Poisson formulas for circular functions on some groups of type H

Explicit Poisson kernels are found for the subelliptic Dirichlet problem with boundary data satisfying certain symmetry conditions on balls and halfspaces in some Heisenberg type groups. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.