On the regularity of solutions to a parabolic system related to Maxwell's equations

Abstract

The goal of this paper is to establish Hölder estimates for the solutions of a certain parabolic system related to Maxwell's equations arising in a quasi-stationary electromagnetic field.  2004 Elsevier Inc. All rights reserved.

Elliptic systems with measurable coefficients of the type of Lamé system in three dimensions

“…Then one can show that u is H ölder continuous in Q T . The proof is very similar to that of [17,Theorem 3.2]. Also, as is mentioned in Remark 3.6, one may assume that a is a 3 × 3 (possibly non-symmetric) matrix valued function satisfying the uniform ellipticity and boundedness condition; see [17] and also consult [18] for treatment of non-symmetric coefficients.…”

“…and prove that weak solutions of the system (7.1) are H ölder continuous in Ω×(0, T) provided that f satisfies some suitable condition, which is an extension of [17,Theorem 3.1], where it is shown that weak solutions of the following system are H ölder continuous:…”

“…There is no essential change in the proof; see [17]. As a matter of fact, one may drop the symmetry assumption on a(x) if one assume further that a ∈ L ∞ (Ω).…”

“…Proof of Theorem 7.3. We follow the strategy used in [17]. As before, we shall make the qualitative assumption that the weak solution u is smooth in Q T .…”

“…Remark 7.5. As in [17,Theorem 3.2], one can consider the case when the coefficients a and b of the system (7.1) are time-dependent but still have some regularity in t-variable. For a measurable function f = f (X) = f (x, t), we set…”