Global regularity for systems with p-structure depending on the symmetric gradient

2.1 Notation

We use c , C to denote generic constants which may change from line to line, but they are independent of the crucial quantities. Moreover, we write f ∼ g if and only if there exists constants c , C > 0 such that c ⁢ f ≤ g ≤ C ⁢ f . In some cases we need to specify the dependence on certain parameters, and consequently we denote by c ⁢ ( ⋅ ) a positive function which is non-decreasing with respect to all its arguments. In particular, we denote by c ⁢ ( δ - 1 ) a possibly critical dependence on the parameter δ as δ → 0 , while c ⁢ ( δ ) only indicates that the constant c depends on δ and will satisfy c ⁢ ( δ ) ≤ c ⁢ ( δ 0 ) for all δ ≤ δ 0 .

We use standard Lebesgue spaces ( L p ( Ω ) , ∥ ⋅ ∥ p ) and Sobolev spaces ( W k , p ( Ω ) , ∥ ⋅ ∥ k , p ) , where Ω ⊂ ℝ 3 is a sufficiently smooth bounded domain. The space W 0 1 , p ⁢ ( Ω ) is the closure of the compactly supported, smooth functions C 0 ∞ ⁢ ( Ω ) in W 1 , p ⁢ ( Ω ) . Thanks to the Poincaré inequality, we equip W 0 1 , p ⁢ ( Ω ) with the gradient norm ∥ ∇ ⋅ ∥ p . When dealing with functions defined only on some open subset G ⊂ Ω , we denote the norm in L p ⁢ ( G ) by ∥ ⋅ ∥ p , G . As usual, we use the symbol ⇀ to denote weak convergence, and → to denote strong convergence. The symbol spt ⁡ f denotes the support of the function f. We do not distinguish between scalar, vector-valued or tensor-valued function spaces. However, we denote vectors by boldface lower-case letter as, e.g., 𝐮 and tensors by boldface upper case letters as, e.g., 𝐒 . For vectors 𝐮 , 𝐯 ∈ ℝ 3 , we denote 𝐮 ⁢ ⊗ 𝑠 ⁢ 𝐯 := 1 2 ⁢ ( 𝐮 ⊗ 𝐯 + ( 𝐮 ⊗ 𝐯 ) ⊤ ) , where the standard tensor product 𝐮 ⊗ 𝐯 ∈ ℝ 3 × 3 is defined as ( 𝐮 ⊗ 𝐯 ) i ⁢ j := u i ⁢ v j . The scalar product of vectors is denoted by 𝐮 ⋅ 𝐯 = ∑ i = 1 3 u i ⁢ v i and the scalar product of tensors is denoted by 𝐀 ⋅ 𝐁 := ∑ i , j = 1 3 A i ⁢ j ⁢ B i ⁢ j .

Greek lower-case letters take only the values 1 , 2 , while Latin lower-case ones take the values 1 , 2 , 3 . We use the summation convention over repeated indices only for Greek lower-case letters, but not for Latin lower-case ones.

2.2 ( p , δ ) -structure

We now define what it means that a tensor field 𝐒 has ( p , δ ) -structure, see [8, 23]. For a tensor 𝐏 ∈ ℝ 3 × 3 , we denote its symmetric part by 𝐏 sym := 1 2 ⁢ ( 𝐏 + 𝐏 ⊤ ) ∈ ℝ sym 3 × 3 := { 𝐏 ∈ ℝ 3 × 3 ∣ 𝐏 = 𝐏 ⊤ } . We use the notation | 𝐏 | 2 = 𝐏 ⋅ 𝐏 ⊤ .

It is convenient to define for t ≥ 0 a special N-function^[2] ϕ ⁢ ( ⋅ ) = ϕ p , δ ⁢ ( ⋅ ) , for p ∈ ( 1 , ∞ ) , δ ≥ 0 , by

The function ϕ satisfies, uniformly in t and independently of δ, the important equivalences

We use the convention that if ϕ ′′ ⁢ ( 0 ) does not exist, the left-hand side in (2.2) is continuously extended by zero for t = 0 . We define the shifted N-functions { ϕ a } a ≥ 0 (cf. [8, 9, 23]), for t ≥ 0 , by

Note that the family { ϕ a } a ≥ 0 satisfies the Δ 2 -condition uniformly with respect to a ≥ 0 , i.e., ϕ a ⁢ ( 2 ⁢ t ) ≤ c ⁢ ( p ) ⁢ ϕ a ⁢ ( t ) holds for all t ≥ 0 .

To a tensor field 𝐒 with ( p , δ ) -structure, we associate the tensor field 𝐅 : ℝ 3 × 3 → ℝ sym 3 × 3 defined through

The connection between 𝐒 , 𝐅 , and { ϕ a } a ≥ 0 is best explained in the following proposition (cf. [8, 23]).

For a detailed discussion of the properties of 𝐒 and 𝐅 and their relation to Orlicz spaces and N-functions, we refer the reader to [23, 3]. Since in the following we shall insert into 𝐒 and 𝐅 only symmetric tensors, we can drop in the above formulas the superscript “ sym ” and restrict the admitted tensors to symmetric ones.

We recall that the following equivalence, which is proved in [3, Lemma 3.8], is valid for all smooth enough symmetric tensor fields 𝐐 ∈ ℝ sym 3 × 3 :

The proof of this equivalence is based on Proposition 2.3. This proposition and the theory of divided differences also imply (cf. [4, Equation (2.26)]) that

for all smooth enough symmetric tensor fields 𝐐 ∈ ℝ sym 3 × 3 .

A crucial observation in [24] is that the quantities in (2.7) are also equivalent to several further quantities. To formulate this precisely, we introduce, for i = 1 , 2 , 3 and for sufficiently smooth symmetric tensor fields 𝐐 , the quantity

Recall that in the definition of 𝒫 i ⁢ ( 𝐐 ) there is no summation convention over the repeated Latin lower-case index i in ∂ i ⁡ 𝐒 ⁢ ( 𝐐 ) ⋅ ∂ i ⁡ 𝐐 . Note that if 𝐒 has ( p , δ ) -structure, then 𝒫 i ⁢ ( 𝐐 ) ≥ 0 for i = 1 , 2 , 3 . The following important equivalences hold, first proved in [24].

2.3 Existence of weak solutions

In this section we define weak solutions of (1.1), recall the main results of existence and uniqueness and discuss a perturbed problem, which is used to justify the computations that follow. From now on, we restrict ourselves to the case p ≤ 2 .

We have the following standard result.

In order to justify some of the following computations, we find it convenient to consider a perturbed problem, where we add to the tensor field 𝐒 with ( p , δ ) -structure a linear perturbation. Using again the theory of monotone operators one can easily prove the following proposition.

2.4 Description and properties of the boundary

We assume that the boundary ∂ ⁡ Ω is of class C 2 , 1 , that is, for each point P ∈ ∂ ⁡ Ω , there are local coordinates such that, in these coordinates, we have P = 0 and ∂ ⁡ Ω is locally described by a C 2 , 1 -function, i.e., there exist R P , R P ′ ∈ ( 0 , ∞ ) , r P ∈ ( 0 , 1 ) and a C 2 , 1 -function a P : B R P 2 ⁢ ( 0 ) → B R P ′ 1 ⁢ ( 0 ) such that

1. 𝐱 ∈ ∂ ⁡ Ω ∩ ( B R P 2 ⁢ ( 0 ) × B R P ′ 1 ⁢ ( 0 ) ) ⇔ x 3 = a P ⁢ ( x 1 , x 2 ) ,

2. Ω P := { ( x , x 3 ) ∣ x = ( x 1 , x 2 ) ⊤ ∈ B R P 2 ⁢ ( 0 ) , a P ⁢ ( x ) < x 3 < a P ⁢ ( x ) + R P ′ } ⊂ Ω ,

3. ∇ ⁡ a P ⁢ ( 0 ) = 𝟎 , and for all x = ( x 1 , x 2 ) ⊤ ∈ B R P 2 ⁢ ( 0 ) , | ∇ ⁡ a P ⁢ ( x ) | < r P ,

where B r k ⁢ ( 0 ) denotes the k-dimensional open ball with center 0 and radius r > 0 . Note that r P can be made arbitrarily small if we make R P small enough. In the sequel, we will also use, for 0 < λ < 1 , the following scaled open sets:

To localize near to ∂ ⁡ Ω ∩ ∂ ⁡ Ω P for P ∈ ∂ ⁡ Ω , we fix smooth functions ξ P : ℝ 3 → ℝ such that

1. χ 1 2 ⁢ Ω P ⁢ ( 𝐱 ) ≤ ξ P ⁢ ( 𝐱 ) ≤ χ 3 4 ⁢ Ω P ⁢ ( 𝐱 ) ,

where χ A ⁢ ( 𝐱 ) is the indicator function of the measurable set A. For the remaining interior estimate, we localize by a smooth function 0 ≤ ξ 00 ≤ 1 , with spt ⁡ ξ 00 ⊂ Ω 00 , where Ω 00 ⊂ Ω is an open set such that dist ⁡ ( ∂ ⁡ Ω 00 , ∂ ⁡ Ω ) > 0 . Since the boundary ∂ ⁡ Ω is compact, we can use an appropriate finite sub-covering which, together with the interior estimate, yields the global estimate.

Let us introduce the tangential derivatives near the boundary. To simplify the notation, we fix P ∈ ∂ ⁡ Ω , h ∈ ( 0 , R P 16 ) , and simply write ξ := ξ P , a := a P . We use the standard notation 𝐱 = ( x ′ , x 3 ) ⊤ and denote by 𝐞 i , i = 1 , 2 , 3 , the canonical orthonormal basis in ℝ 3 . In the following, lower-case Greek letters take the values 1 , 2 . For a function g, with spt ⁡ g ⊂ spt ⁡ ξ , we define, for α = 1 , 2 ,

and if Δ + ⁢ g := g τ - g , we define tangential divided differences by d + ⁢ g := h - 1 ⁢ Δ + ⁢ g . It holds that, if g ∈ W 1 , 1 ⁢ ( Ω ) , then we have for α = 1 , 2

almost everywhere in spt ⁡ ξ (cf. [18, Section 3]). Conversely, uniform L q -bounds for d + ⁢ g imply that ∂ τ ⁡ g belongs to L q ⁢ ( spt ⁡ ξ ) . For simplicity, we denote ∇ ⁡ a := ( ∂ 1 ⁡ a , ∂ 2 ⁡ a , 0 ) ⊤ . The following variant of integration by parts will be often used.