Equality of minimal and maximal extensions of partial differential operators in 𝐿_{𝑝}(𝑅ⁿ)
Ronald Allen Goldstein · Proceedings of the American Mathematical Society · 1966
It is known [l] that if ft is a bounded domain, and P=PiD) is a linear partial differential operator with constant coefficients, then every weak solution in L2(ft) with compact support in ft, is also a strong solution.In this paper, this result is generalized to show that the weak and strong solutions are equivalent for ft = P" and LPiRn); ltkp, without the assumption that the solutions have compact support.We consider a linear partial differential operator of order m, with constant coefficients: P = PiD) = zZ\«\im aaD"; xERn-Here, a -iai, • ■ • , an); the ak are nonnegative integers and |a| = zZak-D = iDu ■ ■ ■ , Dn); Dk = il/i)id/dxk), and D" = D? • ■ ■ D?.The double-barbed arrow "->" will denote strong convergence in Lp, while the single-barbed arrow "-»•" will denote weak convergence