Vol. 30, No. 1, 1969

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Elliptic differential equations with discontinuous coefficients

Donald Platte Squier

Vol. 30 (1969), No. 1, 247–254
Abstract

The purpose of this paper is to furnish a proof of the following theorem:

Theorem. D1 and D2 are two disjoint open sets in the xy-plane having the open arc σ as a common boundary. Li in Di,i = 1,2, are defined as

Li(ϕ) aiϕxx + 2biϕxy + ciϕyy + diϕx
+ eiϕy + giϕ,aici bi2 > 0.
Functions ui satisfy Li(ui) = fi in Di, with ui C2 in Dx and C1 in Dl; on σ,u1 = u2 and ∂u1∕∂N1 = k(s)∂u2∕∂N2, where s is arc length on σ,k(s) > 0, and ∂ui∕∂Ni denotes the conormal derivative of ui. If, on DiUσ,ai,bi,ci Cαn+2;di,ei,gi,fi Cαn; k Cαn+2 and σ Cαn+3; then ui Cn+2 on Difor n 0. If all indicated quantities are analytic functions of their arguments and σ is an analytic arc, then ui is analytic on Di.

Mathematical Subject Classification
Primary: 35.38
Milestones
Received: 19 August 1968
Published: 1 July 1969
Authors
Donald Platte Squier