On the fractional Korn inequality in bounded domains: Counterexamples to the case $ps<1$

The classical Korn’s first inequality [References,References] states that for any vector field $\bm{u}\in W^{1,p}_{0}(\Omega,\mathbb{R}^{n})$ one has

where $\Omega\subset\mathbb{R}^{n}$ is an open bounded Lipschitz domain, $p\in(1,\infty),$ $C>0$ is an absolute constant, and $e(\bm{u})=\frac{1}{2}(\nabla\bm{u}+\nabla\bm{u}^{T})$ is the symmetric part of the gradient. It is also know that if for a field $\bm{u}\in W^{1,p}(\Omega,\mathbb{R}^{n})$ one has $e(\bm{u})=0$ a.e. in $\Omega,$ then $\bm{u}$ must have a constant skew-symmetric gradient [References,References,References]; that is $\bm{u}(\bm{x})=\bm{A}\bm{x}+\bm{b}$ for some $\bm{A}\in\mathbb{R}^{n\times n}$ with $\bm{A}+\bm{A}^{T}=0,$ and some constant vector $\bm{b}\in\mathbb{R}^{n}.$ The fractional analogue of Korn’s first inequality can be formulated as follows. Assuming $s\in(0,1),$ for a vector field $\bm{u}\in L^{p}(\Omega,\mathbb{R}^{n}),$ the seminorm associated to the fractional $s-$derivative of $\bm{u}$ is given by

while the quantity $\|\bm{u}\|_{W^{s,p}(\Omega)}=[\bm{u}]_{W^{s,p}(\Omega)}+\|\bm{u}\|_{L^{p}(\Omega)}$ defines a norm in the space

that is a Banach space [References], see also [References,References]. It then turns out that the fractional analogue of the symmetric part of the $s-$gradient of a field $\bm{u}\in L^{p}(\Omega,\mathbb{R}^{n})$ is given by (following the notation in [References])

It is known that if for a vector field $\bm{u}\in L^{p}(\Omega,\mathbb{R}^{n})$ one has $[\bm{u}]_{\mathcal{X}^{s,p}(\Omega)}=0,$ which is equivalent to $(\bm{u}(\bm{x})-\bm{u}(\bm{y}))\cdot(\bm{x}-\bm{y})=0$ for a.e. $\bm{x},\bm{y}\in\Omega,$ then in fact again $\bm{u}$ has to be an affine map with a skew-symmetric gradient [References, Proposition 1.2], see also [References, Theorem 3,1]. This observation suggests, as also noted by Mengesha [References] and Mengesha and Scott [References], that the fractional analogue of Korn’s first inequality should hold as well:

for all vector fields $\bm{u}\in W^{s,p}_{0}(\Omega,\mathbb{R}^{n}),$ where the space $W^{s,p}_{0}(\Omega,\mathbb{R}^{n})$ is the closure of $C_{c}^{1}(\Omega,\mathbb{R}^{n})$ fields in the $W^{s,p}(\Omega)$ norm. Here the constant $C$ in (1.4) depends only on $n,p,s$ and $\Omega.$ The expression in (1.3) has arisen in the theory of linear peridynamics [References,References,References,References,References,References] as the energy in the small strain regime in the analogy of linear elasticity, where the symmetric gradient $e(\bm{u})$ is the linear strain, and the integral $\int_{\Omega}|e(\bm{u})|^{p}d\bm{x}$ is equivalent to the linear elastic energy. For the purpose of establishing the coercivity of the linear elastic energy, Korn proved [References,References] the estimate (1.1) and the Korn second inequality. The estimate (1.4) was first proven by Mengesha in the case when $\Omega$ is the entire space or the upper half-space $\{x_{n}>0\},$ and $p=2,ps\neq 1$ in [References], and was later extended to any $1<p<\infty,ps\neq 1$ by Mengesha and Scott in [References]. When dealing with non-local operators like on both sides of (1.4), it is sometimes favorable to extend the field $\bm{u}$ into the entire space $\mathbb{R}^{n}$ and work with the extended domain to gain access to Fourier (or other transformation) analysis tools. This strategy has been successfully employed in all of the works [References,References,References], where the idea of Nitsche [References] of extending the field $\bm{u}$ into $\mathbb{R}^{n}$ so that, the symmetric gradient of the extension is suitably controllable plays a vital role. The question of validity of the inequality (1.4) for vector fields $\bm{u}\in W^{s,p}_{0}(\Omega,\mathbb{R}^{n})$ in bounded domains $\Omega$ was addressed in [References,References,References], and has remained open, and is the topic of the present manuscript. We answer this question by proving that in fact (1.4) holds in bounded $C^{1}$-regular domains in the case $ps>1,$ and fails to hold when $ps<1.$ The interesting fact is that in the case $ps<1,$ it does not only fail in general, but it necessarily fails in any open bounded $C^{1}$ domains, see Theorem 2.1. The outcome in the case $ps<1$ in bounded domains is in stark contrast with the case of unbounded domains like the entire space or epigraphs, where (1.4) holds as long as $ps\neq 1$, [References,References,References]. However, it is not surprising as Korn’s first inequality does not hold in the borderline case $p=1$ in the classical case as shown by the celebrated work of Ornstein [References], see also [References] for another approach to constructing counterexamples to such inequalities. For the proof in the case $ps>1,$ following Kondratiev and Oleinik [References] as for the classical case, we employ the recently proven Korn second inequality by Mengesha and Scott [References, Theorem 1.1] in bounded $C^{1}$ domains, which reads as follows:

for all vector fields $\bm{u}\in W_{0}^{s,p}(\Omega,\mathbb{R}^{n}).$ A considerably shorter proof of (1.5) for bounded $C^{1}$ domains or for bounded Lipschitz domains with small Lipschitz constant has been recently given by Rutkowski in [References, Theorem 1.1]

For the case $ps<1,$ we construct the counterexamples $\bm{u}\in C_{c}^{1}(\Omega,\mathbb{R}^{n})$ by interpolating between affine and zero maps, where the nonzero affine value is taken in most of the interior of the domain $\Omega,$ while the zero values are taken near the boundary.

Also, an important consequence of the validity of (1.4) in the case $ps>1$ is that if one extends any given field $\bm{u}\in W_{0}^{s,p}(\Omega,\mathbb{R}^{n})$ onto $\mathbb{R}^{n}$ by zero outside $\Omega,$ then the $W^{s,p}(\mathbb{R}^{n})$ and $\mathcal{X}^{s,p}(\mathbb{R}^{n})$ norms of the extended field $\bar{\bm{u}}$ remain equivalent to the same norms of the original field $\bm{u}$ in $\Omega,$ see Theorem 2.2.

The below theorems contain the main results of the paper.

The next theorem establishes that if one extends any Sobolev field $\bm{u}\in W_{0}^{s,p}(\Omega,\mathbb{R}^{n})$ onto $\mathbb{R}^{n}$ by zero outside $\Omega,$ then the seminorms (in this case those are also norms) $[\bar{\bm{u}}]_{W^{s,p}(\mathbb{R}^{n})}$ and $[\bm{u}]_{W^{s,p}(\Omega)}$ as well as the seminorms $[\bar{\bm{u}}]_{\mathcal{X}^{s,p}(\mathbb{R}^{n})}$ and $[\bm{u}]_{\mathcal{X}^{s,p}(\Omega)}$ will still be equivalent in the case $ps>1.$ This was noted by Rutkowski in [References] under the availability of (2.1). Now, keeping in mind that obviously the norms $\|\bar{\bm{u}}\|_{L^{p}(\mathbb{R}^{n})}$ and $\|\bm{u}\|_{L^{p}(\Omega)}$ are always equivalent, this result would allow one to work with fields defined over $\mathbb{R}^{n}$ instead of $\Omega,$ opening up access to Fourier type analysis in particular.