Related papers: Structure of singularities in the nonlinear nerve …

Considering complex $n$-dimension Calabi-Yau homogeneous hyper-surfaces $% \mathcal{H}_{n}$ with discrete torsion and using Berenstein and Leigh algebraic geometry method, we study Fractional D-branes that result from stringy resolution of…

In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$,…

We consider weak solutions $u:\Omega_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ \Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded open subset of…

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\Delta u=u^{\gamma-1}$, with $\gamma\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. In…

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,\alpha}$ regularity, as well-known in the scalar case \cite{AC,C2}. While in…

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $\Delta u = \lambda_{+}\chi_{\{u>0\}}-\lambda_{-}\chi_{\{u<0\}}, \lambda_\pm>0$. We prove that the free boundary touches the fixed one in…

In this paper, we investigate dead-core problems for fully nonlinear degenerate parabolic equations with strong absorption, \begin{equation*} |Du|^{p} F(D^{2}u) - u_{t} = \lambda_{0}(x,t)\, u^{\mu}\, \chi_{\{u>0\}}(x,t) \qquad \text{in }…

In $n \geq 1$ spatial dimensions, we study the Cauchy problem for a quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology)…

We develop a unified framework for semilinear elliptic equations with gradient-dependent nonlinearities and singular weights in strictly convex domains. Considering large solutions of \[ -\Delta u + b(x)\,h(|\nabla u|) + a(x)\,u = f(x)…

Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of…

In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free…

In this paper, we study semilinear fractional equations $$(-\Delta)^s u(x) = f(u(x))$$ in a half-space and prove that all positive solutions are strictly increasing in the $x_n$-direction. Previous results typically require the solution $u$…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

We consider the Wulff-type energy functional $$ \mathcal{W}_\Omega(u) := \int_\Omega B(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this…

In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^{\alpha}, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n…

In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-\Delta_M, and u locally in…

In this work, we study the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u v \nabla u)-\chi \nabla \cdot\left(u^{2} v \nabla…

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant and we study the behaviour of…

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear elliptic PDEs on the form $$ F(x,u,Du,D^2u) = 0 $$ under suitable structure conditions on the equation allowing for non-Lipschitz growth in…

This paper investigates the regularity of solutions and structural properties of the free boundary for a class of fourth-order elliptic problems with Neumann-type boundary conditions. The singular and degenerate elliptic operators studied…