Optimal regularity for elliptic transmission problems including $C^1$ interfaces

Abstract

We prove an optimal regularity result for elliptic operators −∇ • µ∇ : W 1,q 0 → W −1,q for a q > 3 in the case when the coefficient function µ has a jump across a C 1 interface and is continuous elsewhere. A counterexample shows that the C 1 condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

“…In other words, we require the continuity of the solutions to (SL) for the optimality conditions for (P). In fact, based on maximum elliptic regularity results (see [11,12]), the continuity of the state y is shown in [23]. Hereafter, first-order optimality conditions for (P) were derived.…”

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

“…The question on minimal admissible smoothness of coefficients deserves an additional consideration; cf. [18], [19], [34], and references therein.…”

“…If here n = 2 or 3 and p > 2, then the solutions satisfy the Hölder condition. See, in particular, [18], [19], and references therein, and also Corollary 12.2 in [33]. In our notation, these are the cases of points on the diagonal s + t = 1/2.…”

Регулярность Вариационных Решений Линейных Граничных Задач В Липшицевых Областях

“…Если здесь n = 2 или 3 и p > 2, то решения удовлетворяют условию Гёльдера. См., в частности, [18], [19] и указан-ную там литературу, а также следствие 12.2 в [33]. В наших обозначениях это случаи точек на диагонали s+t = 1/2.…”