Iuliana Marchis
In this paper we study the asymptotic behavior of the solution of quasilinear parametric variational inequalities posed in a cylinder with a thin neck, and we obtain the limit problem.
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We consider arithmetic three-spheres inequalities to solutions of certain second order quasilinear elliptic differential equations and inequalities with a Riccati-type drift term.
We obtain estimates and blow-up conditions for solutions of quasilinear elliptic inequalities containing terms with lower-order derivatives
We investigate the nonexistence and existence of nontrivial positive solutions to $Δ_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to classification of $(p, q)$, we establish different volume growth conditions to obtain Liouville's theorems for the above quasilinear differential inequalities, and we also show these volume growth conditions are sharp in most cases. Moreover, the results are completely new for $(p, q)$ of negative pair, even in the Euclidean space.
This work is devoted to the study of a Liouville comparison principle for entire weak solutions of quasilinear differential inequalities of the form $A(u) + |u|^{q-1}u \leq A(v) + |v|^{q-1}v$ on ${\Bbb R}^n$, where $n\geq 1$, $q$ is positive, and the operator $A(w)$ belongs to a class of the so-called $α$-monotone operators. Typical examples of such operators are the $p$-Laplacian and its well-known modifications. The results improve and supplement those in [1].
We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elli…
Let $f :\R\to\R$ be a continuous function. We prove that under some additional assumptions on $f$ and $A:\R\to\R_{+}$, weak $\Cuno$ solutions of the differential inequality $-\diver(A(\abs{\nabla u})\nabla u)\ge f(u)$ on $\RN$ are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot Groups are considered.
In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $$ Δ_{\mathbb{G}}^φu \ge b(x) f(u) l(|\nabla u|). $$ Prototype examples of $Δ_{\mathbb{G}}^φ$ are the (subelliptic) $p$-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient $l(t)$ that can vanish both as $t \rightarrow 0^+$ and as $t \rightarrow +\infty$. Our results improve on the recent literature and, by means of suitable coun…
In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equati…
We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $ε\ge0$ and $\bar u_ε$ minimizes the functional $$ \int_Ω(ε+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $ψ$, then 0 \le ÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}\bar u_ε|^2)^{(p/2)-1} \nabla_{\H^n}\bar u_ε\Big) \qquad \le (÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}ψ|^2)^{(p/2)-1} \nabla_{\H^n}ψ\Big))^+.
Quasilinear elliptic inequalities with Hardy potential and nonlocal terms
研究带Hardy位势和非局部Riesz位势项的拟线性椭圆不等式，推导得到该不等式在去心单位球外区域正解存在的充要条件。
Let $Ω$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |ζ|^p \le ζ A (x, ζ), \quad |A (x, ζ)| \le C_2 |ζ|^{p-1} $$ with some constants $C_1 > 0$, $C_2 > 0$, and $p > 1$ for almost all $x \in Ω$ and for all $ζ\in {\mathbb R}^n$. We obtain blow-up conditions and priori estimates for inequalities of the form $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge q (x) g (u) \quad \mbox{in } Ω, $$ where $p - 1 \le α\le p$ is a real number and, moreover, $b \in L_{\infty, loc} (Ω)$, $q \in L_{\infty, loc} (Ω)$, and…
In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator
Assume that $p > 1$ and $p - 1 \le α\le p$ are real numbers and $Ω$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge 0, $$ where $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ and $b : Ω\to [0, \infty)$ are some functions with $$ C_1 |ξ|^p \le ξ A (x, ξ), \quad |A (x, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 = const > 0, $$ for almost all $x \in Ω$ and for all $ξ\in {\mathbb R}^n$. For solutions of this inequality we obtain…
We obtain comparison theorems for non-negative solutions of quasilinear elliptic inequalities
Harnack inequalities for quasilinear anisotropic elliptic equations with a first order term
本文针对带一阶项的拟线性各向异性椭圆方程，利用Moser迭代技巧证明了解的Harnack型比较不等式与线性化算子解的Harnack型不等式，并推导出相应的强比较原理和强极大值原理。
Liouville Type Results for Quasilinear Elliptic Inequalities Involving Gradient Terms on Weighted Graphs
本文研究了加权图上的准线性椭圆不等式Δ_m u +u^p |∇u|^q ≤ 0，通过参数范围确定非平凡正解的不存在性，扩展了黎曼流形上的结果，揭示了离散加权图框架下的独特行为和挑战。
On the existence of global solutions of second-order quasilinear elliptic inequalities
研究二阶拟线性椭圆不等式在 ${\mathbb R}^n$ 中全局正解的存在性，在 Carathéodory 函数和结构条件下给出解的存在性结果。
This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In ge…
On global solutions of quasilinear second-order elliptic inequalities
本文研究一类拟线性二阶椭圆不等式，在 Caratheodory 函数满足 p 增长结构条件下，寻找函数 f 的精确条件，以保证任何非负解都恒为零。
We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…
Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces
本文研究一类在非自反Orlicz-Sobolev空间上具有无穷远处消失势的拟线性Schrödinger方程，通过推广Hardy型不等式并结合变分方法，证明了正基态解的存在性，并分析了势函数为零的情形。
On blow-up conditions for solutions of systems of quasilinear second-order elliptic inequalities
本文研究全空间中带 Caratheodory 系数的拟线性二阶椭圆不等式组，刻画其非负解发生爆破的精确条件。
Trace and Hardy-Sobolev type inequalities and applications to a quasilinear elliptic problem in half-space
研究半空间中带权重的拟线性椭圆问题，通过建立新的嵌入不等式和Pohozaev型恒等式，证明弱解的存在性、正则性及非存在性，揭示权重ρ对可解性的关键作用。
We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature operator in non parametric form. We prove that if the exponents lie under a certain curve, then the system has only the trivial solution. These results hold without any restriction provided the possible solutions are more regular. The underlying framework is the classical Euclidean case as well as the Carnot groups setting.
Fujita type results for quasilinear parabolic inequalities with nonlocal terms
本文研究带非局部卷积项的拟线性抛物不等式非负解的不存在性，针对两类带弱m-强制算子的问题分别得到Fujita型临界指数的存在性与不存在性结果，采用适配非局部设定的非线性容量估计方法。
Nonlinear Fisher information, corresponding functional inequalities and applications
该研究分析非线性Fisher信息沿拟线性热方程的演化，提出对应非线性热方程的泛函不等式非线性版本，将其应用于一维Keller--Segel系统得到解的全局存在性，并研究了p-Laplace方程下的Fisher信息版本。
Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms
该研究针对带非局部Riesz位势项的强制拟线性椭圆不等式，推导了正奇异解存在的参数精确条件，并建立了对应双边不等式奇异解的渐近形态。
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in $N-$dimensional domains $Ω$. We also consider singular and degenerate elliptic problems with $A_p$ coefficients involving the $p-$Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the $p-$Laplacian, and compare them with the usual ones in the literature.
Taking only the characteristics as absolute, in the spirit of Arnold's "Geometrical Methods in the Theory of Ordinary Differential Equations" (Springer, 1988), we give an independent of coordinates formulation of general variational entropy inequalities for quasilinear equations of first order, that locally read as Kruzhkov inequalities, in terms of certain "entropy densities", and in the case of the equation of 2D flat projective structure we get the expression of the general entropy density from its abstract Rankine-Hugoniot rule for shocks using the projective geometry of the plane.
We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known that the exponential nonlinearity converges, up to a subsequence, to a sum of Dirac measures. By performing a precise local asymptotic analysis we complete such a result by showing that the corresponding Dirac masses are quantized as multiples of a given one, related to the mass of limiting profiles after rescaling according to the classification result obtain…
The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous noise. In particular, this provides a characterization of solutions to such SPDE in terms of (generalized) strong solutions. Second, for the one-dimensional stochastic mean curvature flow with normal noise we adapt the notion of stochastic variational inequalities to provide a characterization of solutions previously obtained in a limiting sense only. This so…
We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $σ$-gradient ($0<σ<1$). We establish continuous dependence results with respect to the data, including the threshold of the fractional $σ$-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Usin…
The space of Hardy-weights for quasilinear operators on discrete graphs
本文在一般加权图上研究p-Schrödinger算子的Hardy不等式，通过广义容量给出Hardy权空间的Maz'ya型刻画，并利用临界性理论探讨极小化元存在性及其与Cheeger型估计的联系。
Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary
本文研究了具有吸收边界条件的非线性Maxwell系统，证明了小数据解的全局存在性和指数衰减性，并通过迹估计和正则性分析改进了线性自治情况下的结果。
This article is devoted to the study of the behavior of generalized entire solutions for a wide class of quasilinear degenerate inequalities modeled on the following prototype with p-Laplacian in the main part \begin{equation*} {\underset{m}{\overset{i=1}{\sum}}} X_i^*(|\mathbf{X}u|^{p-2} X_i u)\geq |u|^{q-2}u, \ \ x\in {\mathbb{R}}^{n},\ q>1,\ p>1, \end{equation*} where ${\mathbb{R}}^{n}$ is a Carnot-Carathéodory metric space, generated by the system of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes the adjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$…
Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting
本文研究了在黎曼流形上抛物型不等式解的不存在性，通过测试函数方法确定了参数范围，揭示了几何结构、非线性幂次与势函数行为之间的相互作用。
Nontrivial nonnegative weak solutions to fractional $p$-Laplace inequalities
本文研究了分数阶 $p$-Laplace 不等式在局部分数阶 Sobolev 空间中非平凡非负弱解的存在性与不存在性，核心方法是对算子基本解的精细分析。
Asymptotic Stability and the Forcing Term: An Analysis of Non-Newtonian Thin-Film Flows
本文研究了非牛顿薄液膜流动在固态表面的"完全润湿"状态下，通过分析四阶拟线性退化抛物型方程的解的收敛速率，推导出双侧估计并验证了物理现象的一致性。
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