On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

Abstract

We consider the mixed boundary value problem for linear second order elliptic equations in a plane domain lq whose boundary has corners, and obtain conditions sufficient for the solution to be in C2+(), where 0< a < 1. This result means that under those conditions, solutions are as smooth as they would be in the absence of corners, so that, in this sense, the present result is best possible.

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

“…In a subsequent paper, [53], R. Brown and J. Sykes succeeded in proving an L p -version of (1.4), valid for all p ∈ (1,2]. Let us also mention the recent paper [7] by R. Brown, L. Capogna and L. Lanzani in which the authors study the L p Zaremba problem for Laplace's equation in two-dimensional Lipschitz graph domains with Lipschitz constant at most 1.…”

“…Let Ω be an arbitrary, bounded Lipschitz domain in R n and assume that 1 < p 0 , p 1 (3.11) where 0 < θ < 1, s = (1 − θ)s 0 + θs 1 and (3.14) where 0 < θ < 1, s := (1 − θ)s 0 + θs 1 ,…”

“…and we shall call 1) . Similar considerations apply in connection with the real method of interpolation.…”

“…The Poisson problem with mixed Dirichlet-Neumann boundary conditions arises naturally in connection to a series of important problems in mathematical physics and engineering dealing with conductivity, heat transfer, wave phenomena, electrostatics, metallurgical melting, and stamp problems in elasticity and hydrodynamics. Specific references can be found in [1], [12], [13], [17], [22], [23], [27], [33], [36], [42], [43], [46], [47], [48], [49], [50], [57]. Other interesting, physically relevant mixed boundary value problems are those associated with the Maxwell equations and the Dirac operator.…”

The mixed problem in L p for some two-dimensional Lipschitz domains

“…1 We first use a result of Dahlberg and Kenig [10,Theorem 3.8] to reduce to the case where the Dirichlet data in the mixed problem is zero. (While Dahlberg and Kenig only discuss n ≥ 3 in their work, one can extend their results to two dimensions).…”