Related papers: Regularity of the singular set in the fully nonlin…
We consider a system of elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the Laplacian. This system was previously investigated by Caffarelli,…
In the previous work [Interfaces Free Bound., 19, 351--369, 2017], de Queiroz and Shahgholian established the optimal $C^{1,\log}_{\mathrm{loc}}$ regularity of solutions for the obstacle problem with singular logarithmic forcing term…
This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution…
We investigate the local topological structure, stationary point sets in parametric optimization genericly may have. Our main result states that, up to stratified isomorphism, any such structure is already present in the small subclass of…
We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven…
We investigate the regularity of the free boundaries in the 3 elastic membranes problem. We show that the two free boundaries corresponding to the coincidence regions between consecutive membranes are $C^{1,\log}$-hypersurfaces near a…
We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…
We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free…
Given a singular filtration on a manifold, e.g. a subriemannian setting, one can understand the elliptic regularity problems through a special kind of calculus. The principal symbol in this calculus involves the unitary representations of a…
In this paper we give a comprehensive treatment of a two-penalty boundary obstacle problem for a divergence form elliptic operator, motivated by applications to fluid dynamics and thermics. Specifically, we prove existence, uniqueness and…
This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an…
We consider a one-phase free boundary problem involving fractional Laplacian $(-\Delta)^s$, $0<s<1$. D. De Silva, O. Savin, and Y. Sire proved that the flat boundaries are $C^{1,\alpha}$. We raise the regularity to $C^{\infty}$, extending…
We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the…
We prove partial regularity of stationary solutions and minimizers $u$ from a set $\Omega\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_\Omega F(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies…
In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-\Delta)^s +b\cdot\nabla$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free…
We prove that flat or Lipschitz free boundaries of two-phase free boundary problems governed by fully nonlinear uniformly elliptic operators and with non-zero right hand side are $C^{1,\gamma}$.
In this paper we study the higher regularity of the free boundary for the elliptic Signorini problem. By using a partial hodograph-Legendre transformation we show that the regular part of the free boundary is real analytic. The first…
We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider L\'evy operators that are symmetric and its Fourier symbol…
Variational inequalities with thin obstacles and Signorini-type boundary conditions are classical problems in the calculus of variations, arising in numerous applications. In the linear case many refined results are known, while in the…