Dhanya Rajendran | Searcharxiv

arXiv2017

On the boundary conditions in estimating $\nabla ω$ by div $ω$ and curl $ω.$

In this paper we study under what boundary conditions the inequality $$\|\nablaω\|_{L^2(Ω)}^2\leq C\left(\|{\rm curl}ω\|_{L^2(Ω)}^2+ \|{\rm div}ω\|_{L^2(Ω)}^2+\|ω\|_{L^2(Ω)}^2\right) $$ holds true. It is known that such an estimate holds if either the tangential or normal component of $ω$ vanishes on the boundary $\partialω.$ We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions we give an interpolation result between these two classical boundary conditions.

math.AP↗

arXiv2015

Isolated Singularities of Polyharmonic Operator in Even Dimension

We consider the equation $Δ^2 u=g(x,u) \geq 0$ in the sense of distribution in $Ω'=Ω\setminus \{0\} $ where $u$ and $ -Δu\geq 0.$ Then it is known that $u$ solves $Δ^2 u=g(x,u)+αδ_0-βΔδ_0,$ for some non-negative constants $α$ and $ β.$ In this paper we study the existence of singular solutions to $Δ^2 u= a(x) f(u)+αδ_0-βΔδ_0$ in a domain $Ω\subset \mathbb{R}^4,$ $ a$ is a non-negative measurable function in some Lebesgue space. If $Δ^2 u=a(x)f(u)$ in $Ω',$ then we find the growth of the nonlinearity $f$ that determines $α$ and $β$ to be $0.$ In case when $α=β=0,$ we will establish regularity results when $f(t)\leq C e^{γt},$ for some $C, γ>0.$ This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions $(N\geq 5)$ with a specific weight function $a(x)=|x|^σ.$ Later we discuss its analogous generalization for the polyharmonic operator.

math.AP↗