Related papers: A $C^1$ regularity result for the inhomogeneous no…

We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly…

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f $ in dimension two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is $C^{2,\alpha}$…

In this article, we establish global regularity results ($ C^{0,\gamma}$, $ C^{0,1} $ and $ C^{1}$ estimates) for a class of degenerate fully nonlinear equation on $ C^{2} $-domain. This corresponds to the boundary counterpart of the…

The general exact solution of the Einstein-Dirac equations with cosmological constant in the homogeneous Riemannian space of the Bianchi 1 type is obtained.

We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds under appropriate assumptions.

We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate…

We prove $L^p(b\mathcal{D})$-regularity of the Cauchy-Leray integral for bounded domains $\mathcal{D}\subset\mathbb C^n$ whose boundary satisfies the minimal regularity condition of class $C^{1,1}$, together with a naturally occurring…

In this paper, we obtain $C^{1}$ and $C^{1,1}$ regularity of $L^{n}$-viscosity solutions for general semilinear elliptic equation in nondivergence form under some more weaker assumptions, which generalize the result for equations with…

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive…

We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order $m$ ($1 \leq m \leq n$) with a continuous right hand side and a continuous boundary data in a bounded…

We establish an optimal C^{1,\alpha}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.

We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \begin{equation*} \left\{\begin{aligned} \Ds u&= \lambda_{s, p}…

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla…

We introduce a notion of convexity with respect to a one-dimensional operator and with this notion find a one-parameter family of different convexities that interpolates between classical convexity and quasiconvexity. We show that, for this…

On a bounded domain $\Omega$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies H\"ormander's bracket generating…

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$,…

We consider the normalized $p$-Poisson problem $$-\Delta^N_p u=f \qquad \text{in}\quad \Omega.$$ The normalized $p$-Laplacian $\Delta_p^{N}u:=|D u|^{2-p}\Delta_p u$ is in non-divergence form and arises for example from stochastic games. We…

In the present paper, we study sharp C^{1;\alpha} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.

We present a result of existence of infinitely many solutions for the Dirichlet problem involving the p-Laplacian in annular domains, when $p\leq N$, contouring the failure of compactness of $W^{1,p}(\Omega)$ in $C^0(\bar{\Omega})$ applying…