Elliptic differential equations with discontinuous coefficients

Donald Platte Squier

Vol. 30 (1969), No. 1, 247–254

DOI: 10.2140/pjm.1969.30.247

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. L_i in D_i,i = 1,2, are defined as

Li(ϕ)	≡ aiϕxx + 2biϕxy + ciϕyy + diϕx
	+ eiϕy + giϕ,aici − bi2 > 0.

Functions u_i satisfy L_i(u_i) = f_i in D_i, with u_i ∈ C² in D_x and ∈ C¹ in D_lUσ; on σ,u₁ = u₂ and ∂u₁∕∂N₁ = k(s)∂u₂∕∂N₂, where s is arc length on σ,k(s) > 0, and ∂u_i∕∂N_i denotes the conormal derivative of u_i. If, on D_iUσ,a_i,b_i,c_i ∈ C_αⁿ⁺²;d_i,e_i,g_i,f_i ∈ C_αⁿ; k ∈ C_αⁿ⁺² and σ ∈ C_αⁿ⁺³; then u_i ∈ Cⁿ⁺² on D_iUσ for n ≧ 0. If all indicated quantities are analytic functions of their arguments and σ is an analytic arc, then u_i is analytic on D_iUσ.

Mathematical Subject Classification

Primary: 35.38

Milestones

Received: 19 August 1968

Published: 1 July 1969