Related papers: Viscosity solutions to uniformly elliptic complex …
We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1, \alpha}$,…
In this paper, we study the cone degenerate p-Laplace equation. We provide the existence of the viscosity solutions by proving Alexandrov-Bakelman-Pucci and H\"older estimates. Further more, we give the comparison principle by an equivalent…
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations…
In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient…
We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior…
In this survey we prove H\"older regularity results for viscosity solutions of fully nonlinear nonlocal uniformly elliptic second order differential equations with local gradient terms. This extends the nonlocal counterpart of the work of…
In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable…
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for…
We prove the local gradient H\"older regularity of viscosity solutions to the inhomogeneous normalized $p(x)$-Laplace equation $$ -\Delta u-(p(x)-2)\frac{\left\langle D^{2}uDu,Du\right\rangle }{\left|Du\right|^{2}} = f(x), $$ where $p$ is…
We establish that when n >= 2 and H is a C^1 Hamiltonian such that some level set contains a line segment, the Aronsson equation admits explicit entire viscosity solutions. They are superpositions of a linear part plus a Lipschitz…
We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.
This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is…
In this work we provide an Aleksandrov-Bakelman-Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a…
We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with…
In this paper we prove a H\"older regularity estimate for viscosity solutions of inhomogeneous equations governed by the infinite Laplace operator relative to a frame of vector fields.
In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by $$ \left\{ \begin{array}{rcl} F(D^2u,x) &=& f(x) \quad…
In this paper we prove the H\"older regularity of bounded, uniformly continuous, viscosity solutions of some degenerate fully nonlinear equations in the Heisenberg group.
We prove a quantitative H\"{o}lder continuity result for viscosity solutions to the equation $$ (-\Delta_p)^{s}u(x) + {\rm PV} \int_{\mathbb{R}^n} |u(x)-u(x+z)|^{q-2}(u(x)-u(x+z))\frac{\xi(x,z)}{|z|^{n+ tq}} dz=f \quad \text{in}\; B_2, $$…
We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first, we establish…
We study boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations. The gradient-dependent degeneracy or singularity, along with the time derivative, introduces significant challenges beyond…