Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,…,uN) : EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,inX,on∂X, where X is a finite, open, symmetric N -annulus (with N≥2 ), P=P(x) is an unknown hydrostatic pressure field and φ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N≥4 and discuss a number of closely related issues

On a class of stationary loops on SO(n) and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

For full abstract please refer to http://dx.doi.org/10.1186/s13661-018-1047-

On a class of (p,q)(p,q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

summary:Let $\Omega \subset \mathbb{R}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem \begin{align*} -\Delta_p u-\Delta_q u = \lambda ({\bf x} )\lvert u\rvert^{p^\star -2} u+\mu |u|^{r-2} u \end{align*} where $\mu$ is a positive parameter, $1 < q \le p < n$, $r\ge p^{\star}$ and $p^{\star}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity

On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem -∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u where μ is a positive parameter, 1 &lt; q ≤ p &lt; n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity

On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian proble

On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian proble

Generalised twists, SO(n) and the p-energy over a space of measure preserving maps

Let View the MathML source be a bounded Lipschitz domain and consider the energy functional View the MathML source with pset membership, variant]1,8[ over the space of measure preserving maps View the MathML source In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler¿Lagrange equations associated with View the MathML source over View the MathML source. The main result is a surprising discrepancy between even and odd dimensions. Here we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler¿Lagrange equations where a key ingredient is a necessary and sufficient condition for an associated vector field to be a gradient