We give the infinite-dimensional representation for the ellipticK-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic K-matrix associated to Belavin’s completely Z-symmetric R-matrix. PACS: ¶ arch-ive/9507123 † [email protected] 1 The quantum R-matrix as solutions of the Yang-Baxter equ...

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special cases are analyzed. In conclusion, for completeness, a concrete example of an application is presented by finding the existence of three nontrivial weak soluti...

Optimal relaxation parameters are obtained for red-black Gauss-Seidel relaxation in multigrid solvers of a family of elliptic equations. The resulting relaxation schemes are found to retain very high ef-ciency over an appreciable range of coeecients of the elliptic differential operator, yielding simple, inexpensive and fully paralleliz-able smoothers in many situations where less cost-eeective...

In this article, we will consider second order uniformly elliptic operators of divergence form defined on Rn with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x1, . . . , xn}, a second order uniformly elliptic operator of divergence form, L, ...

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels. Indiana Univ. Ma...

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space Wm 2 (G) are studied. The Fredholm property of the unbounded operator corresponding to the elliptic equation, acting on L2(G), and defined for functions from the space Wm 2 (G) that satis...

In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Holder regularity of the weak solutions follows in a standard way. 1 Let Sf = YJl ,=i 9j(ajjdj) be a second-order degenerate ellipt...

We present a two-scale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We consider the elliptic problem in mixed variational form over W × V ⊂ L2 × H(div). We base...

In this paper, we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives. Four types of basic spectral regularization methods such as Showalter’s, Tikhonov’s, Lardy’s and Lavrentiev’s are applied to approximate the solution by introducing another large (or small) parameter. Convergence of the regularized solutions to t...

Parenthetically, we observe that if L is strongly elliptic, then L− λ, λ ∈ R, satisfies the non-singularity hypothesis (3) relative to any subdomain Ω ⊆ M provided λ is sufficiently large. This is a consequence of Garding’s inequality, which is valid in our setting (even though V may be unbounded). Also, clearly, if L is strongly elliptic and negative semidefinite, then L−λ satisfies (3) for an...