Related papers: On a weighted two-phase boundary obstacle problem
In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with…
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free…
In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation}…
In this paper we study a double-phase problem involving the 1-Laplacian with non-homogeneous Dirichlet boundary conditions and show the existence and uniqueness of a solution in a suitable weak sense. We also provide a variational…
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $\Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of…
In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schr\"odinger operators. We deduce the strong unique continuation property in the presence of subcritical and…
Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at…
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue…
In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $\mathbb{R}^N$. The equation is driven by the fractional Laplacian $(-\Delta)^{\frac{s}{2}}$…
In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…
In this paper, we explore cooperative and competitive coupled obstacle systems, which, up to now, are new type obstacle systems and formed by coupling two equations belonging to classical obstacle problem. On one hand, applying the…
We study the parabolic obstacle problem $$\lap u-u_t=f\chi_{\{u>0\}}, \quad u\geq 0,\quad f\in L^p \quad \mbox{with}\quad f(0)=1$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular…
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace…
We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a…
In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(\Delta)_{p}^{s} u + V(x)|u|^{p-2}u &= \lambda f(u),&x\in\Omega; u&=0,&x\in…
By employing a new class of pseudo-differential operators introduced in a previous work, we establish a novel monotonicity formula for the fractional Laplacian $|\nabla_x|^\alpha$ in $\mathbb{R}^n$, with $n \geq 2$ and $\alpha \in [1,2)$,…
In this paper we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic…
In this paper, we develop a sliding method for the fractional Laplacian. We first obtain the key ingredients needed in the sliding method either in a bounded domain or in the whole space, such as narrow region principles and maximum…
We study the existence, uniqueness, and regularity of weak solutions to a class of obstacle problems, where the obstacle condition can be imposed on a subset of the domain. In particular, we establish the optimal H\"older regularity for…
A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain…