L p -trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

For $1< p<\infty$ we prove an $L^{p}$ -version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \] holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ , i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$ . We also show the norm equivalence \begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*} for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ . These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

Positive radial solutions for a noncooperative resonant nuclear reactor model with sign-changing nonlinearities

This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system: $$ \textstyle\begin{cases} -\Delta u=uv+f( \vert x \vert ,u), & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ -\Delta v=cg(u)-dv, & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ \frac{\partial u}{\partial \textbf{n}}=0= \frac{\partial v}{\partial \textbf{n}},& \vert x \vert =R_{1}, \vert x \vert =R_{2}, \end{cases} $$ { − Δ u = u v + f ( | x | , u ) , 0 < R 1 < | x | < R 2 , x ∈ R N , − Δ v = c g ( u ) − d v , 0 < R 1 < | x | < R 2 , x ∈ R N , ∂ u ∂ n = 0 = ∂ v ∂ n , | x | = R 1 , | x | = R 2 , where $\mathbb{R}^{N}$ R N ($N\geq 1$ N ≥ 1 ) is the usual Euclidean space, n indicates the outward unit normal vector, $f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})$ f ∈ C ( [ R 1 , R 2 ] × [ 0 , ∞ ) , R ) , $g\in C([0,\infty ),[0,\infty ))$ g ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

Using the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

Patterns in a generalized volume-filling chemotaxis model with cell proliferation

In this paper, we consider the following system [Formula: see text] which corresponds to the stationary system of a generalized volume-filling chemotaxis model with logistic source in a bounded domain in [Formula: see text] with zero Neumann boundary conditions. Here the parameters [Formula: see text] are positive and [Formula: see text], and [Formula: see text] denotes the outward unit normal vector of [Formula: see text]. With the priori positive lower- and upper-bound solutions derived by the Moser iteration technique and maximum principle, we apply the degree index theory in an annulus to show that if the chemotactic coefficient [Formula: see text] is suitably large, the system with [Formula: see text] admits pattern solutions under certain conditions. Numerical simulations of the pattern formation are shown to illustrate the theoretical results and predict the interesting phenomenon for further studies.

Spaces and Operators

This chapter consists mainly of definitions and various properties (without proofs) of spaces and operators used in this book. It defines O as an open set in Rᶰ such that it is locally on one side of its boundary Γ‎ := δ‎O, which is supposed to be bounded and Lipschitz. The chapter is mainly focused on the case of N = 3. Further, without loss of generality, the chapter supposes that Γ‎ is connected (for otherwise, one could work separately at each connected component). Such a set O is referred to as ‘regular’ in what follows. Let n denote the outward unit normal vector to Γ‎. In addition, let Oₑ := Rᶰ∖Ō: By N₀ we denote the set N ∪ {0}.

Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion

With the help of the I-integrals expressed by Mura (1987, Micromechanics of Defects in Solids, 2nd ed., Martinus Nijhoff, Dordrecht) and the outward unit normal vector introduced by Ju and Sun (1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574), the closed form solution of the exterior-point Eshelby tensor for an elliptic cylindrical inclusion is derived in this work. The proposed closed form of the Eshelby tensor for an elliptic cylindrical inclusion is more explicit than that given by Mura, which is rough and unfinished. The Eshelby tensor for an elliptic cylindrical inclusion can be reduced to the Eshelby tensor for a circular cylindrical inclusion by letting the aspect ratio of the inclusion α=1. The closed form Eshelby tensor presented in this study can contribute to micromechanics-based analysis of composites with elliptic cylindrical inclusions.