A Simple Proof of Regularity for C1,α Interface Transmission Problems

Hongjie Dong

Division of Applied Mathematics, Brown University, 182 George Street,Providence, RI 02912, USA

Abstract. We give an alternative proof of a recent result in [1] by Caffarelli,Soria-Carro, and Stinga about the C1,α regularity of weak solutions to transmission problems with C1,α interfaces. Our proof does not use the mean value property or the maximum principle, and also works for more general elliptic systems with variable coefficients. This answers a question raised in [1]. Some extensions to C1,Dini interfaces and to domains with multiple sub-domains are also discussed.

Key words: Transmission problems, elliptic systems with variable coefficients, C1,α and C1,Dini interfacial boundaries, Cα and Dini coefficients.

1 Introduction and main results

In a recent paper[1],Caffarelli, Soria-Carro,and Stinga studied the following transmission problem. Let ?∈Rbe a smooth bounded domain with d≥2, and ?be a sub-domain of ? such that ???? and ?=??. Assume that the interfacial boundary Γ(=??) between ?and ?is Cfor some α ∈(0,1). Consider the elliptic problem with the transmission conditions where g is a given function on Γ, ν is the unit normal vector on Γ which is pointing inside ?, and u|and u|(and ?u|and ?u|) are the left and right limit of u(and its normal derivative, respectively) on Γ in ?and ?. The main result of [1]can be formulated as the following theorem.

Theorem 1.1. Under the assumptions above, for any g ∈C(Γ), there is a unique weak solution u∈H(?) to (1.1), which is piecewise Cup to the boundary in ?and ?and satisfies

where N=N(d,α,?,Γ)>0 is a constant.

The proof in [1] uses the mean value property for harmonic functions and the maximum principle together with an approximation argument. We refer the reader to [1] for earlier results about the transmission problem with smooth interfacial boundaries. The main feature of Theorem 1.1 is that Γ is only assumed to be in C, which is weaker than those in the literature. In Remark 4.5 of[1], the authors raised the question of transmission problems with variable coefficient operator and mentioned two main difficulties in carrying over their proof to the general case.

In this paper, we answer this question by giving a alternative proof of Theorem 1.1, which does not invoke the mean value property or the maximum principle, and also works for more general non-homogeneous elliptic systems in the form

where the Einstein summation convention in repeated indices is used,

are(column)vector-valued functions,for k,l=1,···,d,A=A(x)are n×n matrices,which are bounded and satisfy the strong ellipticity with ellipticity constant κ>0:

Theorem 1.2. Assume that ?, ?, and Γ satisfy the conditions in Theorem 1.1,Aand F are piecewise Cin ?and ?, g∈C(Γ),and f ∈L(?). Then there is a unique weak solution u∈H(?) to (1.2), which is piecewise Cup to the boundary in ?and ?and satisfies

where N=N(d,n,κ,α,?,Γ,[A])>0 is a constant.

We also consider the transmission problem with multiple disjoint sub-domains?,···,?with Cinterfacial boundaries in the setting of[5,6]. As in these papers,we assume that any point x∈? belongs to the boundaries of at most two of the ?s,so that if the boundaries of two ?touch, then they touch on a whole component of such a boundary. Without loss of generality assume that ????, j=1,···,M?1 and ?????. The transmission problem in this case is then given by

In the following theorem,we obtain an estimate which is independent of the distance of interfacial boundaries, but may depend on the number of sub-domains M.

Theorem 1.3. Assume that ?satisfy the conditions above,Aand F are piecewise Cfor some α∈(0,α/(1+α)], g∈C(??), j=1,···,M?1, and f ∈L(?). Then there is a unique weak solution u∈H(?) to (1.3), which is piecewise Cup to the boundary in ?, j=1,···,M, and satisfies

where N=N(d,n,M,κ,α,α,?,[A])>0 is a constant.

It is worth noting that in the special case when Aand F are H¨older continuous in the whole domain,by the linearity the result of Theorem 1.3 still holds with α=α.

Our last result concerns the case when the interfaces are C, and Asatisfy the piecewise Dini mean oscillation in ?, i.e., the function

satisfies the Dini condition, where ?(x)=B(x)∩? and A is the set of piecewise constant functions in ?, j=1,···,M.

Theorem 1.4. Assume that ?satisfy the Ccondition, Aand F are of piecewise Dini mean oscillation in ?, gis Dini continuous on ??, j=1,···,M ?1, and f ∈L(?). Then there is a unique weak solution u∈H(?) to (1.3), which is piecewise Cup to the boundary in ?, j=1,···,M.

We note that the piecewise Dini mean oscillation condition is weaker than the usual piecewise Dini continuity condition in the Lsense.

2 Proofs

The idea of the proof is to reduce the transmission problem to an elliptic equation (system) with piecewise H¨older (or Dini)non-homogeneous terms, by solving a conormal boundary value problem. These equations arose from composite material and have been extensively studied in the literature. See, for instance, [5,6], and also recent papers [2,4]. We will apply the results in the latter two papers, the proofs of which in turn are based on Campanato’s approach.

Proof of Theorem 1.2. Let w∈H(?)be the weak solution to the conormal boundary value problem

where N =N(d,?). Since g ∈C(?), by the classical elliptic theory (see, for instance, [7, Theorem 5.1]), we have

where N=N(d,α,?). By using the weak formulation of solutions, from (2.1) it is easily seen that (1.2) is equivalent to

The theorem is proved.

Proof of Theorem 1.3. The proof is similar to that of Theorem 1.2. In each ?,j=1,···,M ?1, we find a weak solution to

where

and wsatisfies

By using the weak formulation of solutions, it is easily seen that (1.3) is equivalent to

where As before, by the Lax–Milgram theorem, there is a unique solution u ∈H(?) to(2.8). Since ?F and Aare piecewise Cand ??is piecewise C, by using (2.7)and appealing to [4, Corollary 1.2 and Remark 1.4], we conclude the proof of the theorem.

Finally, we give

Proof of Theorem 1.4. We claim that under the conditions of the theorem, if wis the solution to (2.6), then Dwsatisfies the L-Dini mean oscillation condition in?. Assuming this is true, then the conclusion of the theorem follows from the proof of Theorem 1.3 and [4, Theorem 1.1]. We remark that the Ccontinuity of wwas proved in [7, Theorem 5.1] for more general quasilinear equations, but in general Dwmay not be Dini continuous in the Lsense.

To prove the claim, we follow the argument in the proof of Theorem 1.7 of [3].We only give the boundary estimate since the corresponding interior estimate is simpler. By using the Cregularity of ?and locally flattening the boundary, it then suffices to verify Lemma 2.1 below.

In the sequel,we denote x=(x,x),where x=(x,x,···,x)∈R,and Γ(x):=B(x)∩{x=0} for x∈Rand r>0. We say that a function f satisfies the L-Dini mean oscillation in ? if

satisfies the Dini condition.

Acknowledgements

H. Dong was partially supported by the Simons Foundation, grant No. 709545.