Related papers: Lyapunov-Type Inequalities for Third Order Nonline…
We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities was studied in…
Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive…
In this brief note we discuss local H\"older continuity for solutions to anisotropic elliptic equations of the type $ \sum_{i=1}^s \partial_{ii} u+ \sum_{i=s+1}^N \partial_i \bigg(A_i(x,u,\nabla u) \bigg) =0,$ for $x \in \Omega \subset…
In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
In this paper, we study the following quasi-linear elliptic inequality $\Delta_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m,…
Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws…
We consider fully nonlinear degenerate elliptic equations with zero and first order terms. We provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which…
In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\M^+(D^2u)+|u|^{s-1}u=f(x) \quad {in} \RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This…
This article deals with the study of the following nonlinear doubly nonlocal equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+\ba(-\Delta)^{s_2}_{q}u = \la a(x)|u|^{\delta-2}u+ b(x)|u|^{r-2} u,\; \text{ in }\; \Om, \; u=0 \text{ on }…
We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational…
We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions…
This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form $$-div (|x|^{a} D u ) = f(x,u), ~ u > 0,\, \mbox{ in } \Omega,$$ where $N \geq 3$, $\Omega$ is an open domain in…
This paper considers a sampling-based approach to stability verification for piecewise continuous nonlinear systems via Lyapunov functions. Depending on the system dynamics, the candidate Lyapunov function and the set of initial states of…
We study a class of boundary value problems with $\varphi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\varphi(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u')\right)'=\lambda f(u)\; \text{ on }(-L,…
In this paper, we develop a unified framework able to certify both exponential and subexponential convergence rates for a wide range of iterative first-order optimization algorithms. To this end, we construct a family of parameter-dependent…
In this article was proved, that any linear equation $Lu=f$ in the case of any linear versions of the Galerkin method, has at least as many unsolved right-hand sides in the form of linear combinations $f=L\psi_1 +...+L\psi_N +L\psi_{N+1}…
In this paper, we obtain some new inequalities of Hermite-Hadamard type and Simpson type for functions whose third derivatives belong to Godunova-Levin class.
It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -\Delta_\Phi u = \lambda a(x) |u|^{q-2}u + |u|^{p-2}u, & x\in\Omega, u = 0, & x \in \partial…
We consider a class of stationary Schr\"{o}dinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schr\"{o}dinger operator $-\Delta+V$ is indefinite. Previous results in this framework…