Related papers: The $L_p$-Gaussian Minkowski problem
We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$…
We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of B\"or\"oczky, Lutwak, Yang and Zhang, and the Dimensional…
We obtain inequalities of H\"{o}lder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of non-negative weights.
Certain excess versions of the Minkowski and H\"older inequalities are given. These new results generalize and improve the Minkowski and H\"older inequalities.
We extend the results of [5], where we proved an equivalence between weighted Poincar\'e inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar…
It is shown that for a given Banach space $X$ and a subspace $Y$ weakly $\mathcal{K}$-analytic, $L_p(\mu,Y)$ is $p$-simultaneously proximinal in $L_p(\mu,X)$ whenever $Y$ is $p$-simultaneously proximinal in $X$.
We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the…
We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to $n$ independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also…
In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern \cite{Ch59}, we obtain the…
The Banach-Saks property is an important tool in analysis with applications ranging from partial differential equations (PDEs) to calculus of variations and probability theory. We survey the Banach-Saks property for $L^p$-spaces, with a…
In this paper we introduce a new class of Gaussian singular integrals, the general alternative Gaussian singular integrals and study the boundedness of them in Lp, p >1 and its weak (1,1) boundedness with respect to the Gaussian measure…
We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…
In this paper, we study $C^{1, 1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem. Our proof is motivated by the idea of Guan and Li's work on $C^{1,1}$ estimates for solutions to the Aleksandrov problem.
We prove existence of weak solutions to the Cauchy problem corresponding to various strictly parabolic equations on a compact Riemannian manifold $(M,g)$. This also includes strictly parabolic equations with stochastic forcing with linear…
The goal of this paper is to solve backward doubly stochastic differential equation (BDSDE, in short) under weak assumptions on the data. The first part is devoted to the development of some new technical aspects of stochastic calculus…
In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -\Delta _{p(\cdot )}u=f(u)-\lambda & \text{in }\Omega \\ u>0 &…
In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is established and some criteria on local smoothness and wave…
In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff-Schr\"{o}dinger-Poisson system involving the weak singularity, i.e. when $0<\gamma<1$. Further, we obtain the existence of a solution with the…
This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem \ref{general-dual-polar}). This problem involves the general dual volume $\widetilde{V}_G(\cdot)$ recently proposed in \cite{GHWXY, GHXY}…