In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f&\text{in }\Omega, u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.
We establish the interior $C^{1,α}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^γF(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{γ/2}F(D^2u) &&\text{i...
We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(|x|^\beta(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=\frac{|u|^{q-2}u}{|x|^\alpha} \ \ \text{ in }\Omega, \end{equation} where $2\leq p0$ are fixed exponents, $\Omega$ is an open subset o...
We study boundary regularity for the fully nonlinear singular elliptic equation of the form $|Du|^\gamma F(D^2u)=f$, where $-1<\gamma<0$. First, we prove pointwise boundary $C^{1,\alpha}$ estimates with pointwise $C^{1,\alpha}$ Dirichlet data, which weakens the $C^2$ domain smoothness requirement from Birindelli and De...
We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \begin{equation*} \Delta u+V(x)u=0\qquad\text{in }\Omega, \qquad u=0\qquad\text{on }\partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\...
This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $01$ and $...
For a (not necessarily smooth) bounded domain Ω$\Omega$ of RN$\mathbb {R}^N$ , N⩾2$N \geqslant 2$ and a Carathéodory vector‐valued function a:Ω×RN→RN$a:\Omega \times \mathbb {R}^N \rightarrow \mathbb {R}^N$ , we study the compactness of the inverse of the Leray–Lions operator A(u)=−div(a(x,∇u))$A(u)=-\text{div}(a(x, \n...
D. Arcoya, M. C. Rezende, E. A. Silva· Journal of the London Mathem...· 0 citations
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