The asymptotically sharp Korn interpolation and second inequalities for shells

# The asymptotically sharp Korn interpolation and second inequalities for shells

D. Harutyunyan
###### Abstract

We consider shells in three dimensional Euclidean space which have bounded principal curvatures. We prove Korn’s interpolation (or the so called first and a half111The inequality first introduced in [References]) and second inequalities on that kind of shells for vector fields, imposing no boundary or normalization conditions on The constants in the estimates are optimal in terms of the asymptotics in the shell thickness having the scalings or The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré type estimate with the symmetrized gradient on the right hand side. In particular this applies to linear geometric rigidity estimates for shells, i.e., Korn’s fist inequality without boundary conditions.

## 1 Introduction

A shell of thickness in three dimensional Euclidean space is given by where is a bounded and connected smooth enough regular surface with a unit normal at the point The surface is called the mid-surface of the shell Understanding the rigidity of a shell is one of the challenges in nonlinear elasticity, where there are still many open questions. Unlike the situation for shells in general, the rigidity of plates has been quite well understood by Friesecke, James and Müller in their celebrated papers [References,References]. It is known that the rigidity of a shell is closely related to the optimal Korn’s constant in the nonlinear (in some cases linear) first Korn’s inequality [References,References], which is a geometric rigidity estimate for fields [References,References,References,References,References]. Depending on the problem, the field may or may not satisfy boundary conditions, e.g. [References,References,References]. Finding the optimal constants in Korn’s inequalities is a central task in problems concerning shells in general. The Friesecke-James-Müller estimate reads as follows: Assume is open bounded connected and Lipschitz. Then there exists a constant such that for every vector field there exists a constant rotation , such that

 ∥∇u−R∥2≤CI∫Ωdist2(∇u(x),SO(3))dx. (1.1)

The linearization of (1.1) around the identity matrix is Korn’s first inequality [References,References,References,References,References] without boundary conditions and reads as follows: Assume is open bounded connected and Lipschitz. Then there exists a constant depending only on such that for every vector field there exists a skew-symmetric matrix i.e., such that

 ∥∇u−A∥2L2(Ω)≤CII∥e(u)∥2L2(Ω), (1.2)

where is the symmetrized gradient (the strain in linear elasticity). The estimate (1.2) is traditionally proven by using Korn’s second inequality, that reads as follows: Assume is open bounded connected and Lipschitz. Then there exists a constant depending only on such that for every vector field there holds:

 ∥∇u∥2L2(Ω)≤C(∥u∥2L2(Ω)+∥e(u)∥2L2(Ω)). (1.3)

It is known that if is a thin domain with thickness then in general the optimal constants in all inequalities (1.1)-(1.3) blow up as In particular, if is a plate given by where is open bounded connected and Lipschitz, then as proven in [References] one has and asymptotically as While the asymptotics of is known in the case when satisfies zero Dirichlet boundary conditions on the thin face of the shell [References,References] ( scaling like or ), it is open for general fields In this work we are concerned with the asymptotics of the constant in (1.3) or more precisely in the so called Korn interpolation inequality, or the first-and-a-half Korn inequality [References], in the general case when is a shell. The statements solving the problem practically completely appear in the next section.

## 2 Main Results

We first introduce the main notation and definitions. We will assume throughout this work that the mid-surface of the shell is connected, compact, regular and of class up to its boundary. We also assume that has a finite atlas of patches such that each patch can be parametrized by the principal variables and (constant and constant are the principal lines on ) that change in the ranges for where for Moreover, the functions and satisfy the conditions

 min1≤i≤kinfθ∈[0,ωi][z2i(θ)−z1i(θ)]=l>0,max1≤i≤ksupθ∈[0,ωi][z2i(θ)−z1i(θ)]=L<∞, (2.1) max1≤i≤k(∥z1i∥W1,∞[0,ωi]+∥z2i∥W1,∞[0,ωi])=Z<∞.

Since there will be no condition imposed on the vector field (see Theorem 2.1), we can restrict ourselves to a single patch and denote it by for simplicity. If the parametrization of is and is the unit normal to denoting the normal variable by and we get

 ∇u=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ut,tut,θ−AθκθuθAθ(1+tκθ)ut,z−AzκzuzAz(1+tκz)uθ,tAzuθ,θ+AzAθκθut+Aθ,zuzAzAθ(1+tκθ)Aθuθ,z−Az,θuzAzAθ(1+tκz)uz,tAzuz,θ−Aθ,zuθAzAθ(1+tκθ)Aθuz,z+AzAθκzut+Az,θuθAzAθ(1+tκz)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ (2.2)

in the orthonormal local basis where and are the two principal curvatures. Here we use the notation for the partial derivative inside the gradient matrix of a vector field The gradient on or the so called simplified gradient denoted by is obtained from (2.2) by putting We will work with and then pass to using their closeness to the order of due to the smallness of the variable In this paper all norms are norms and the inner product of two functions will be given by which gives rise to the norm . In what follows in the below theorems, the constants and will depend only on the shell mid-surface parameters, which are the quantities and where Our results are Korn’s interpolation and second inequalities for the shell providing sharp Ansatz-free lower bounds for displacements imposing no boundary condition on the field The estimates are also proven to be asymptotically optimal as

###### Theorem 2.1 (Korn’s interpolation inequality).

There exists constants such that Korn’s interpolation inequality holds:

 ∥∇u∥2≤C(∥u⋅n∥⋅∥e(u)∥h+∥u∥2+∥e(u)∥2), (2.3)

for all and where is the unit normal to the mid-surface Moreover, the exponent of in the inequality (2.3) is optimal for any shell satisfying the above imposed regularity condition together with (2.1), i.e., there exists a displacement realizing the asymptotics of in (2.3).

###### Theorem 2.2 (Korn’s second inequality).

We get by the Cauchy-Schwartz inequality from (2.3) the following Korn’s second inequality for shells: There exists constants such that Korn’s second inequality holds:

 ∥∇u∥2≤Ch(∥u∥2+∥e(u)∥2), (2.4)

for all and Moreover, the exponent of in the inequality (2.4) is optimal for any shell satisfying the above imposed regularity condition together with (2.1), i.e., there exists a displacement realizing the asymptotics of in (2.4).

## 3 The key lemma

In this section we prove a gradient separation estimate for harmonic functions in two dimensional thin rectangles, which is one of the key estimates in the proof of Theorem 2.1.

###### Lemma 3.1.

Assume such that Denote There exists a universal constat such that any harmonic function fulfills the inequality

 ∥wy∥2L2(Rb)≤C(1h∥w∥L2(Rb)⋅∥wx∥L2(Rb)+1b2∥w∥2L2(Rb)+∥wx∥2L2(Rb)). (3.1)
###### Sketch of proof.

We divide the proof into four steps for the convenience of the reader. Let us point out that all the norms in the proof are unless specified otherwise.
Step 1. An estimate on rectangles. Assume and denote There exists a universal constat such that any harmonic function fulfills the inequality

 ∥wy−a∥L2(R)≤ch∥wx∥L2(R), (3.2)

where is the average of over the rectangle Estimate (3.2) is derived from the linear version of (1.1) for plates, i.e., the estimate (1.2) for as mentioned in the previous section. Indeed, considering the plate and the displacement one gets (3.2) with instead of but the quantity is minimized at Therefore (3.2) follows.
Step 2. An interior estimate on There exists an absolute constant such that for any harmonic function the inequality holds:

 ∫(h/4,3h/4)×(0,b)|wy|2≤C(1h∥w∥⋅∥wx∥+1b2∥w∥2+∥wx∥2). (3.3)

Let be a parameter and let be a smooth cutoff function such that for and for Next for we denote and We multiply the equality in by and integrate the obtained identity first by parts over and then in over to get the estimate

 ∫Rh/4,z|∇w|2≤4h∫Rb|wwx|+1ϵ2z2∫Rbotz∪Rtopzw2+ϵ2∫Rbotz∪Rtopzw2y, (3.4)

where is a parameter yet to be chosen. By the invariance of (3.2) under the variable change we have for some

 ∫Rbot2z|wy−a1|2≤cz2h2∫Rbot2z|wx|2,and∫Rtop2z|wy−a2|2≤cz2h2∫Rtop2z|wx|2, (3.5)

which gives together with the triangle inequality the estimates

 ∫Rh/4,z|∇w|2≥hz4(a21+a22)−cz2h2∫Rbot2z|wx|2−cz2h2∫Rtop2z|wx|2. (3.6)

An application of the triangle inequality to in (3.4) and utilization of (3.5) and (3.6) derives from (3.4) for the value the estimate

 hz8(a21+a22)≤4h∫Rb|wwx|+16z2∫Rbotz∪Rtopzw2+2cz2h2∫Rbot2z∪Rtop2z|wx|2. (3.7)

Newt we combine (3.4) (for ), (3.5) and (3.7) to get the key interior estimate

 ∫Rh/4,0|wy|2≤C(1h∫Rb|wwx|+1z2∥w∥2+z2h2∥wx∥2). (3.8)

It remains to minimize the right hand side of (3.8) subject to the constraint on the parameter to get (3.3) The procedure is standard and is left to the reader.
Step 3. An estimate near the horizontal boundary of . There exists an absolute constant such that for any harmonic function the inequality holds:

 ∫Rboth∪Rtoph|wy|2≤C(1h∫Rb|wwx|+1b2∥w∥2+∥wx∥2). (3.9)

The proof is similar to Step1 by the utilization of (3.5) and (3.7).
Step 4. Proof of (3.1). We recall the following two auxiliary lemmas proven by Kondratiev and Oleinik [References], see also [LABEL:bib:Harutyunyan.1].

###### Lemma 3.2.

Assume and is absolutely continuous. Then the inequality holds:

 ∫a0f2(t)dt≤4∫2aaf2(t)dt+4∫2a0t2t′2(t)dt. (3.10)
###### Lemma 3.3.

Let and let be open bounded connected and Lipschitz. Denote Assume is harmonic. Then there holds:

 ∥δ∇u∥L2(Ω)≤2∥∇u∥L2(Ω). (3.11)

Fixing a point and applying Lemma 3.2 to the function on the segment as a function in we get

 ∫(0,h/4)×(h,b−h)|wy|2≤∫(h/4,h/2)×(h,b−h)|wy|2+4∫(0,h/2)×(h,b−h)|xwxy|2. (3.12)

Lemma 3.3 applied to the harmonic function reduces (3.12) to the key estimate

 ∫(0,h/4)×(h,b−h)|wy|2≤∫(h/4,h/2)×(h,b−h)|wy|2+16∫Rb|wx|2. (3.13)

It remains to combine a similar estimate for the right part of the rectangle with (3.13), (3.9) and (3.3). ∎

## 4 Proof of the main results

###### Sketch of proof of Theorem 2.1.

Let us point out that throughout this section the constants will depend only on the quantities and unless specified otherwise. We first prove the estimate with and in place of and in (2.3), which we do block by block by freezing each of the variables and .
The block . We aim to prove the estimate

 ∥F23∥2+∥F32∥2≤C(∥u∥2+∥e(F)∥2). (4.1)

Denote and assume satisfies the conditions for all Then, for any displacement considering the auxiliary vector field one can get from Korn’s second inequality [References], that there exists a constant depending only on the constants and such that for the matrix fulfills the estimate

 ∥Mφ∥2L2(Rt)≤c(∥e(Mφ)∥2L2(Rt)+∥u∥2L2(Rt)+∥v∥2L2(Rt)). (4.2)

An application of (4.2) for and gives (4.1). We combine the estimates for the other two blocks in one by first proving the following Korn-like inequality on thin rectangles, which will be the key estimate for the rest of the proof.

###### Lemma 4.1.

For denote Given a displacement the vector fields and the function denote the perturbed gradient as follows:

 M=[uxuy+α⋅Uvxvy+β⋅U+w]. (4.3)

Assume then the following Korn-like interpolation inequality holds:

 ∥M∥2L2(R)≤C(∥u∥L2(R)⋅∥e(M)∥L2(R)h+∥e(M)∥2L2(R)+1ϵ∥U∥2L2(R)+ϵ(∥wL2(R)∥2+∥wx∥2L2(R))), (4.4)

for all small enough, where depends only on the quantities and

###### Proof.

Let us point out that in the proof of Lemma 4.1, the constant may depend only on and as well as the norm will be First of all, we can assume by density that For functions denote Assume is the harmonic part of in i.e., it is the unique solution of the Dirichlet boundary value problem

 {Δ~u(x,y)=0,(x,y)∈R~u(x,y)=u(x,y),(x,y)∈∂R. (4.5)

The Poincaré inequality gives the bound Multiplying the identity by we get by the Schwartz inequality the bounds

 ∥∇(u−~u)∥≤C[∥e(Mf,g)∥+h(∥fy∥+∥gx∥)],∥u−~u∥≤Ch[∥e(Mf,g)∥+h(∥fy∥+∥gx∥)]. (4.6)

In the next step we utilize the fact that is harmonic, thus we can apply Lemma 3.1 to First apply the triangle inequality to get and then we apply Lemma 3.1 to the summand first and then the triangle inequality several times (also taking into account the bounds (4.6)) to get the estimate

 ∥uy+f∥2 ≤C(1h∥u∥⋅∥e(Mf,g)∥+∥u∥(∥fy∥+∥gx∥)+∥u∥2+∥e(Mf,g)∥2+∥f∥2). (4.7)

For the special case and one has the bounds and thus an application of the Cauchy-Schwartz inequality (involving the parameter ) leads (4.7) to (4.4). ∎

The block . For the block we freeze the variable and deal with two-variable functions. We aim to prove that for any the estimate holds:

 ∥F13∥2+∥F31∥2≤C(∥ut∥⋅∥e(F)∥h+∥e(F)∥2+1ϵ∥u∥2+ϵ∥F21∥2), (4.8)

where the norms are over the whole shell

###### Proof.

Indeed, it is not difficult to see that (4.8) follows from Lemma 4.1 with the following choice: Fix and consider the displacement the vector fields and the function in the variables and over the thin rectangle

The block . The role of the variables and is the completely the same, thus we have an analogous estimate

 ∥F12∥2+∥F21∥2≤C(∥ut∥⋅∥e(F)∥h+∥e(F)∥2+1ϵ∥u∥2+ϵ∥F31∥2). (4.9)

Consequently adding (4.8) and (4.9) and choosing the parameter small enough we discover

 ∥F12∥2+∥F21∥2+∥F13∥2+∥F31∥2≤C(∥ut∥⋅∥e(F)∥h+∥e(F)∥2+∥u∥2). (4.10)

A combination of (4.1) and (4.10) completes the proof of the lower bound. It remains to note that one gets (2.1) from that with in place of by an application of the obvious bounds and The Ansatz realising the asymptotics of in (2.3) and (2.4) has been constructed in [References] and reads as follows:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ut=W(θ√h,z)uθ=−t⋅W,θ(θ√h,z)Aθ√huz=−t⋅W,z(θ√h,z)Az, (4.11)

where is a smooth and periodic in function that the derivative is not identically zero. The calculation is omitted here. ∎

## References

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