A product property of Sobolev spaces with application to elliptic estimates

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p 2 [1;1), n;m 2 Z‡, and m 2 that satisfy m > n=p, kfckm;p;V K sup Vs jfj ! kckm;p;V ‡ kckmÿ1;q;V ‡ kckmÿ1;p;V kfkm;p;V " # for all f;c 2 W(V) that satisfy sptc Vs V and domains V R that are nonempty, open, and satisfy the cone condition. Here q ˆ p if p > n, q 2 (n= ; pn=(nÿ p)] if n > p, q 2 (n= ;1) if p ˆ n, K ˆ K(n; p;m; q; C), where C is the cone from the cone condition, and :ˆ [[ n=p ]], the largest integer less than or equal to n=p. MATHEMATICS SUBJECT CLASSIFICATION (2010). 46E35, 35J57, 74B15. KEYWORDS. Elasticity, elliptic regularity, Sobolev estimate, systems of partial differential equations. 1. Introduction; Sobolev Spaces A standard classical methodology used to obtain a priori estimates for elliptic systems of partial differential equations is to first prove the required estimate when the system has constant coefficients and the region has smooth boundary and then use a partition of unity to extend the es(*) Indirizzo dell'A.: Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA. E-mail: hsimpson@math.utk.edu (**) Indirizzo dell'A.: Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA. E-mail: sspector@siu.edu timate to coefficients that depend on position and regions that are less regular. For example, Agmon, Douglis, and Nirenberg [3, 4] first establish the estimate (in the notation from Elasticity): for all u 2 C1(V; R) that satisfy u ˆ 0 in a neighborhood of D :ˆ @VnS, kukm‡1;p;V N Div C[ru] mÿ1;p;V ‡ C[ru]n mÿ1 p;p;S ‡ kukp;V ; …1:1† where C : M n ! M n is a constant linear mapping of the n n matrices M n and V is a ball with S ˆ [ or V is a half-ball and S is the flat portion of the boundary of V. Here m 2 Z‡, p 2 (1;1), n is the outward unit normal to the boundary @V,