Related papers: Bounded solution to Hessian type equations for $(\…
Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of…
In this paper, we prove a Brezis-Merle type inequality for $k$-convex functions vanishing on the boundary. As an application, we establish an Alexandrov-Bakelman-Pucci type estimate for the intermediate Hessian equation. Furthermore, we…
In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a P\'olya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of…
This paper studies the Neumann boundary value problem for sum Hessian equations. We first derive a priori $C^2$ estimates for $(k-1)$-admissible solutions in almost convex and uniformly $(k-1)$-convex domains, and prove the existence of…
In a previous paper \cite{ref1} we produced a sequence of analytic functions $\{\alpha \uparrow^n z\}_{n=0}^\infty$ when $1 \le \alpha \le e^{1/e}$ and $z$ was in the right half of the complex plane, the \emph{bounded analytic…
This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-\Delta)^{\frac{\alpha}{2}}u + u - C_{n,-\beta}…
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…
In this paper, we study the existence of positive entire large and bounded radial positive solutions for a nonlinear system. Our results give an answer of the question raised in [11].
We prove upper and lower bounds for a variational functional for convex functions satisfying certain boundary conditions on a sector of the unit ball in two dimensions. The functional contains two terms: The full Hessian and its…
Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb B$ of the $n$-dimensional complex Euclidean space $\mathbb C^n$ such that the function $f$ vanishes on the set ${\sf Z}\subset \mathbb B$ and satisfies the constraint…
Given a set of inequalities determined by homogeneous forms, the following intertwined results are established: (1) the volume of the real semi-algebraic domain determined by these inequalities is explicitly determined; it is shown to be…
In this paper, we establish the existence of large solutions of Hessian equations and obtain a new boundary asymptotic behavior of solutions.
In this paper, we used some theorems of fixed point for studying the results of existence and uniqueness for Hilfer-Hadamard-Type fractional differential equations, \[_{H}D^{\alpha,\beta}x(t)+f(t,x(t))=0, \hbox{ on the interval } J:=(1,e]\]…
This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$ using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield…
In this paper, we prove the symmetry of the solution to overdetermined problem for the equation $\sigma_k(D^2u-uI)=C_n^k$ in hyperbolic space. Our approach is based on establishing a Rellich-Pohozaev type identity and using a P function.…
In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we…
This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\Delta{}u^{m}$. Our main objective is to improve the H$\ddot{o}$lder estimate obtained by pioneers…
Some years ago, Fendley found an explicit solution to Thermodynamic Bethe Ansatz (TBA) equation for a N=2 supersymmetric theory in 2D with a specific F-term. Motivated by this, we seek for explicit solutions for other super-potential cases…
We study the solutions of the wave equation where a massless scalar field is coupled to the Wahlquist metric, a type-D solution. We first take the full metric and then write simplifications of the metric by taking some of the constants in…
In this paper, we investigate the boundary H\"{o}lder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic…