Related papers: Smooth local solutions to weingarten equations and…

Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, \alpha}(\overline{\Omega}) $ estimates for strictly $(\delta, \widetilde{\gamma}_k)…

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in H\"older-Zygmund spaces $C^r(\mathbb{R}^2)$ for $r>1$ and uniformly local Sobolev spaces…

We study the Seiberg-Witten equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric which is "asymptotic" to a Poincar\'e type metric at…

In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…

We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a…

We study the regularity of the viscosity solution $u$ of the $\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $\Omega$. We prove…

We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain…

All real solutions of the Lane-Emden equation for n = 5 are obtained in terms of Jacobian and Weierstrass elliptic functions. A new family of solutions is found. It is expressed by remarkably simple formulae involving Jacobian elliptic…

We study the distribution of solutions of equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$. We give new upper bounds for these solutions.

In this paper, we prove global-in-time existence and uniqueness of smooth solutions to the homogeneous Landau-Fermi-Dirac equation with Coulomb potential. The initial conditions are nonnegative, bounded and integrable. We also show that any…

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group.…

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $\Omega\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical…

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated…

In this short note, smoothness of the fundamental solution of Schr\"odinger equations on a complete manifold is studied. It is shown that (1) the fundamental solution is smooth under "mild" trapping conditions; (2) there is a Riemannian…

We show that the there exists an algebraic rate of convergence of solutions of finite-difference approximations for uniformly elliptic Isaacs in smooth bounded domains.

This paper is concerned with existence and qualitative properties of positive solutions of semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We show the existence of positive solutions in the vicinity of…

This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

The purpose of this paper is to establish local regularity of the solution operator to the Kohn-Laplace equation, called the complex Green operator, on abstract CR manifolds of hypersurface type. For a cut-off function $\sigma$, we…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…