In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $$M^n$$ with $$n\ge 2$$ immersed in the Euclidean unit ball $$\mathbb {B}^{n+k}$$ under one of pinching conditions $$|\Phi |^2\le C$$ , $$|A|^2\le {\widetilde{C}}$$ or |\le R(p,|H|)$$ where A $$(\Phi )$$ is (traceless) second fundamental form, H mean curvature, ...

Abstract We study the dynamics of a population with an age structure whose range expands time, where adult is assumed to satisfy reaction–diffusion equation over changing interval determined by Stefan type free boundary condition, while juvenile satisfies evolving domain population. The interactions between and populations involve fixed time-delay, which renders model nonlocal in nature. After ...

We study a nonlinear generalization of free boundary problem that arises in the context thermal insulation. consider two open sets $\Omega\subseteq A$, and we search for an optimal $A$ order to minimize non-linear energy functional, whose minimizers $u$ satisfy following conditions: $\Delta_p u=0$ inside $A\setminus\Omega$, $u=1$ $\Omega$, Robin-like $(p,q)$-condition on $\partial A$. variation...

In free-boundary problems, the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems that include both sources of error. The derivation of an appropriate dual problem (linearized adjoint) is, ...