Non-unique solutions to boundary value problems for non-symmetric divergence form equations

Abstract

Abstract. We calculate explicitly solutions to the Dirichlet and Neumann boundary value problems in the upper half plane, for a family of divergence form equations having non-symmetric coefficients with a jump discontinuity. It is shown that the boundary equation method and the Lax-Milgram method for constructing solutions may give two different solutions when the coefficients are sufficiently non-symmetric.

Solvability of elliptic systems with square integrable boundary data

“…Clearly, (7) implies (5), which in turn implies (8), as is seen by taking f (x)=∇ t,x g(t, x) for fixed t and then integrating over t. Furthermore (8), implies (6), which is seen by taking g(t, x):=ψ(εt)u 0 (x) and integrating away t, for some ψ∈C ∞ 0 (R + ). Letting ε→∞ and ε→0, respectively, proves (6).…”

“…On the other hand, if A is of block form, i.e. A 0 =A 0 =0, then (5) is equivalent to (6) and to the R n+1 + Gårding inequality (8), for H 1 (R n+1 + ; C m ) as well as for H 1 0 (R n+1 + ; C m ), since (6) implies (5). It is also known that the R n+1 + Gårding inequality (8) implies strong accretivity (7) when m=1, so for scalar equations (5), (8) and (7) are all equivalent.…”

“…As pointed out in Section 2, the standing assumption that A is strictly accretive on N(curl ), i.e. (5), implies that the Gårding inequality (8) (20) below defines a bounded functional onḢ 1 (R 1+n + ; C m ) if φ=div x w, where φ∈L 2 (R n ; C m ) and w∈L 2 (R n ; C nm ), since R n |φ(ξ)| 2 max(|ξ| −1 , 1) dξ <∞ and the trace map V →v mapsḢ 1 (R 1+n…”

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

“…Note that since ᐄ ⊂ L 2 ‫ޒ(‬ + × S n ; ᐂ), solutions to the regularity and Neumann problem always coincide with the variational solutions, by the uniqueness of such. In the setting of the half-space, as in [Auscher et al 2010b] and [Part I], it was shown in [Axelsson 2010] that this uniqueness result does not hold. As pointed out in [Auscher et al 2010b, Remark 5.6], the problem occurs at infinity for the regularity and Neumann problems, which explains why uniqueness holds for the bounded ball.…”

“…As pointed out in [Auscher et al 2010b, Remark 5.6], the problem occurs at infinity for the regularity and Neumann problems, which explains why uniqueness holds for the bounded ball. Although the analogue of [Axelsson 2010] for the Dirichlet problem on the ball is not properly understood at the moment, Theorem 19.4 below shows that uniqueness of solutions essentially holds also for the Dirichlet problem on the unit ball.…”