In this article we use a flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible domain, we derive the C 1,α-regularity of the free boundary in a small ball centred at the origin. From the C 1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally C 3,α up to the free boundary, and therefore C 2,1. In the end we study an example, showing that in general C 2, 1 2 is the best regularity that a solution may achieve in dimension n ≥ 2.

In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstaclesarising in the optimal switching problem with two modes. We derive the optimal C 1,1 -regularity for the minimal solution under the assumption that the zero loop set L := {ψ 1 + ψ 2 = 0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C 1,1 -regularity does not hold without the assumption L = L 0 .

In this paper we study the following parabolic system $$\begin{aligned} \Delta \mathbf{u }-\partial _t \mathbf{u }=|\mathbf{u }|^{q-1}\mathbf{u }\,\chi _{\{ |\mathbf{u }|>0 \}}, \qquad \mathbf{u }= (u^1, \cdots , u^m) \ , \end{aligned}$$ Δ u - ∂ t u = | u | q - 1 u χ { | u | > 0 } , u = ( u 1 , ⋯ , u m ) , with free boundary $$\partial \{|\mathbf{u }| >0\}$$ ∂ { | u | > 0 } . For $$0\le q<1$$ 0 ≤ q < 1 , we prove optimal growth rate for solutions $$\mathbf{u }$$ u to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $$C^{1, \alpha }$$ C 1 , α in space directions and half-Lipschitz in the time direction.

In this article we study a normalised double obstacle problem with polynomial obstacles p 1 ≤ p 2 under the assumption that p 1 (x) = p 2 (x) iff x = 0. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials p 1 , p 2 . In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets {u = p 1 } and {u = p 2 } are cones with a common vertex.We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particular we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then locally the free boundary consists of four C 1,γ -curves, meeting at the origin.In the end we give an example of a three-dimensional double-cone solution.