We study the regularity of free boundary in parabolic obstacle problem for fractional Laplacian $$(-\Delta )^s$$ (and more general integro-differential operators) regime $$s>\frac{1}{2}$$ . prove that once is $$C^1$$ it actually $$C^{2,\alpha }$$ To do so, we establish a Harnack inequality and $$C^{1,\alpha (moving) domains, providing quotient two solutions linear equation, vanish on boundary, ...

We obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite Prandtl number with rigid lower boundary and stress-free upper boundary. Because of the asymmetric boundary conditions the solutions of the Euler-Lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on th...

Given a countable group $G$ splitting as free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the $Out(G,\mathcal{F})$ all outer automorphisms that preserve conjugacy classes each $G_i$. We show every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains relatively fully irreducible automorphism, or else it virtually preserves...

Abstract Throughout this section, we will use the notation $$\displaystyle W_0(u)=\int _{B_1}|\nabla u|{ }^2\,dx-\int _{\partial B_1}u^2\,d\mathcal {H}^{d-1}\qquad \text{and}\qquad W(u)=W_0(u)+|\{u>0\}\cap B_1|, $$ W 0 ( u...