Optimal Regularity of Mixed Dirichlet-Conormal Boundary Value Problems for Parabolic Operators

Mixed boundary value problems for parabolic equations in Sobolev spaces with mixed-norms

“…In this paper, we extend the result in [4] by proving L t q (L x p ) mixed-norm estimates for Du and the H 1 q,p (see (2.1)) solvability when…”

“…p solvability and localization. In this section, we derive some local estimates, in the proofs of which, we shall use the unmixed-norm L p -estimates proved in [4]. For the reader's convenience, we present here the main result in [4].…”

“…In this process, standard techniques including the mollification and iteration arguments as in the proof of [4,Lemma 4.3] are needed. Here we omit the details.…”

“…The proof in [4] relies on a decomposition argument using a carefully designed cut-off function near the separation Γ and on a level set method with the measure theoretic "crawling of ink spots" lemma originally due to Krylov and Safonov [23,18]. While in [4] we used the decomposition argument and estimates in L 2based spaces, in this paper, to prove our main result, we refine the decomposition argument in [4] in the setting of L p -based spaces for p < 2 and exploit an idea of Krylov [17] to utilize the level set method in the t-variable only. Because the decomposition argument fails for p > 2, it remains open whether the mixed-norm estimates hold for…”

“…See [12]. In a recent paper [4], we proved the unique solvability in unmixed-norm Sobolev spaces H 1 p (see (2.1) and (2.2)) for the problem (1.1) when the coefficients a i j have small bounded mean oscillation (BMO) with respect to all the variables (t, x), the base domain Ω is Reifenberg-flat, and the separation Γ is locally close to the graph of a Lipschitz function of m variables, where m ∈ {0, 1, . .…”