Related papers: Concavity properties for solutions to $p$-Laplace …

The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a…

We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional…

We consider a class of fully non-linear parabolic equations on compact Hermitian manifolds involving symmetric functions of partial Laplacians. Under fairly general assumptions, we show the long time existence and convergence of solutions.…

We investigate the existence of nonnegative solutions for a nonlinear problem involving the fractional p-Laplacian operator. The problem is set on a unbounded domain, and compactness issues have to be handled.

We prove that the critical problem for the $p$-Laplacian operator admits a nontrivial solution in annular shaped domains with sufficiently small inner hole. This extends Coron's problem to a class of quasilinear problems.

Let $\Omega$ be a strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^{k+2}$ boundary, $k \geq 1$. We construct a $\overline\partial$ solution operator (depending on $k$) that gains $\frac12$ derivative in the Sobolev space $H^{s,p}…

In this paper we prove the existence of a weak solution to a doubly nonlinear parabolic fractional $p$-Laplacian equation, which has general doubly non-linearlity including not only the Sobolev subcritical/critical/supercritical cases but…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

Matrix versions of some basic convexity inequalities are given. Further results on the same topic are proved in the recent papers on arxiv: 1. Hermitian operators and convex functions, 2. A concavity inequality for symmetric norms, 3.…

We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative…

We obtain new $L_p$ estimates for subsolutions to fully nonlinear equations. Based on our $L_p$ estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations,…

Under appropriate assumptions, we show that all bounded entire solutions to a class of semilinear elliptic systems are confined in a convex domain. Moreover, we prove a Liouville type theorem in the case where the domain is strictly convex.…

In this paper, we establish some new Hadamard type inequalities using elementary well known inequalities for functions whose inequalities absolute values are {\alpha}-, m-, ({\alpha},m)-logarithmically convex.

We construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for…

Sobolev-type regularity results are proved for solutions to a class of second order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak…

In this paper, we study properties of solutions of the Dominative $p$-Laplace equation with homogeneous Dirichlet boundary conditions in a bounded convex domain $\Omega$. For the equation $-\mathcal{D}_p u= 1$, we show that $\sqrt{u}$ is…

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open…

We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher…

We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand…

In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we investigate the shape of solutions whose derivative in $\theta$ vanishes at the…