Abstract We consider the vectorial analogue of thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization nonlocal version classical Bernoulli problem. study optimal regularity, nondegeneracy, and density properties local minimizers. Via blow-up analysis based on Weiss type monotonicity formula, we show that is union “regular” “singular” part...

In this article, we study the behavior of two species evolving in a domain with free boundary. This system mimics spread invasive or new predator species, which boundaries represent expanding fronts and are described by Stefan condition. A priori estimates for required functions established. These used to prove existence uniqueness solution.

Abstract This chapter is dedicated to the measure theoretic structure of free boundary ∂ Ω u . The results presented here are mainly a consequence Lipschitz continuity and non-degeneracy minimizer (Theorem 3.1 Proposition 4.1 ).

Abstract We consider the local existence and uniqueness of solutions for a system consisting an inviscid fluid with free boundary, modeled by Euler equations, in domain enclosed elastic which evolves according to wave equation. derive priori estimates also conclude uniqueness. Both, are obtained under assumption that data belongs $$H^{r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML...