Related papers: Pogorelov type $C^2$ estimates for sum Hessian equ…
We consider the problem \begin{equation}\label{Eq:Abstract} (1)\;\;\;\begin{cases} S_k(D^2u)= \lambda (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit…
We show that the equation $\lambda_1 n_1^2 + ... + \lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, provided that $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy…
In this paper, we derive an \emph{a priori} second order estimate for solutions which are in $\Gamma_{k+1}$ cone to a class of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian…
We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2…
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show…
A Chebyshev-type quadrature for a probability measure sigma is a distribution which is uniform on n points and has the same first k moments as sigma. We give an upper bound for the minimal n required to achieve a given degree k, for sigma…
We study the existence of $L^2$ normalized solutions for nonlinear Schr\"odinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \,…
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…
In this paper, we prove the existence of normalized solutions for the following Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u-\lambda u=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right.…
This paper is devoted to various applications of Hardy-Sobolev type inequalities. We derive a new $L^2$ estimate for the $\bar{\partial}-$equation on ${\mathbb C}^n$ which yields a quantitative generalization of the Hartogs extension…
In this paper we propose some approaches for finding of pointwise estimates of a solution of the Dirichlet boundary value problem $-\Delta u \pm |u|^{q-1} u = 0 $, $|u|=k$ when $|x|=d<1$ and $|u|=0$ when $|x|=1$ where $x\in \Omega = \{x|…
We consider sums of Hurwitz class numbers of the type $\sum_{t \equiv m \pmod{M}}H(4p-t^2)$, where $M$ is composite. For $M=6$ and $8$, we show that these sums can be expressed in terms of coefficients of CM cusp forms. This leads to…
In recent years two Krylov subspace methods have been proposed for solving skew symmetric linear systems, one based on the minimum residual condition, the other on the Galerkin condition. We give new, algorithm-independent proofs that in…
We establish some $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $\Sigma$ as a…
We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this…
We establish interior $W^{2,\delta}$ type estimates for a class of degenerate fully nonlinear elliptic equations with $L^n$ data. The main idea of our approach is to slide $C^{1,\alpha}$ cones, instead of paraboloids, vertically to touch…
This paper investigates the existence of a global $C^{1,1}$ solution to the Dirichlet problem for the $k$-Hessian equation with a nonnegative right-hand side $f$, focusing on the required conditions for $f$. The conditions $f^{1/(k-1)}\in…
In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Amp\`{e}re type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in…
In this paper, we study the $\bar\partial$-equation on some convex domains of infinite type in $\mathbb C^2$. In detail, we prove that supnorm estimates hold for infinite exponential type domains provided the exponent is less than 1.
In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order…