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Yehuda Pinchover

Publications and source records attributed to Yehuda Pinchover.

50 records · Page 3Linked to original sources
arXiv2010

Positive Liouville theorems and asymptotic behavior for p-Laplacian type elliptic equations with a Fuchsian potential

We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u):= - pLaplace(u) + V |u|^{p-2} u = 0 in X, where X is a domain in R^d, d > 1, and 1<p<infty. We assume that the potential V has a Fuchsian type singularity at a point zeta, where either zeta=infty and X is a truncated C^2-cone, or zeta=0 and zeta is either an isolated point of a boundary of X or belongs to a C^2-portion of the boundary of X.

math.AP↗
arXiv2010

On the Hardy-Sobolev-Maz'ya inequality and its generalizations

The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.

math.AP↗
arXiv2009

On positive solutions of p-Laplacian-type equations

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_Ω. (|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.

math.AP↗
arXiv2007

On positive solutions of minimal growth for singular p-Laplacian with potential term

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by Q(u):=\frac{1}{p}\int_Ω(|\nabla u|^p+V|u|^p)\dx, Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u. It is assumed that $Q\geq 0$ on $C_0^\infty(Ω)$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^\infty(Ω)$ and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.

math.AP↗
arXiv2007

A Liouville-type theorem for the p-Laplacian with potential term

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p-Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2), this condition involves comparison of both the functions and of their gradients.

math.AP↗
arXiv2006

Spectral properties of Schrodinger operators defined on N-dimensional infinite trees

We study the discreteness of the spectrum of Schrodinger operators which are defined on N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.

math.SP↗
arXiv2006

A Liouville-type theorem for Schrödinger operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator $P_1$, such that a nonzero subsolution of a symmetric nonnegative operator $P_0$ is a ground state. Particularly, if $P_j:=-Δ+V_j$, for $j=0,1$, are two nonnegative Schrödinger operators defined on $Ω\subseteq \mathbb{R}^d$ such that $P_1$ is critical in $Ω$ with a ground state $ϕ$, the function $ψ\nleq 0$ is a subsolution of the equation $P_0u=0$ in $Ω$ and satisfies $|ψ|\leq Cϕ$ in $Ω$, then $P_0$ is critical in $Ω$ and $ψ$ is its ground state. In particular, $ψ$ is (up to a multiplicative constant) the unique positive supersolution of the equation $P_0u=0$ in $Ω$. Similar results hold for general symmetric operators, and also on Riemannian manifolds.

math.AP↗
arXiv2005

Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds

The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at the edges of the spectrum. Here one says that the Liouville theorem holds if the space of solutions of any given polynomial growth is finite dimensional. The necessary and sufficient condition for a Liouville type theorem to hold is that the real Fermi surface of the elliptic operator consists of finitely many points (modulo the reciprocal lattice). Thus, such a theorem generically is expected to hold at the edges of the spectrum. The precise description of the spaces of polynomially growing solutions depends upon a `homogenized' constant coefficient operator determined by the analytic structure of the dispersion relation. In most cases, simple explicit formulas are found for the dimensions of the spaces of polynomially growing solutions in terms of the dispersion curves. The role of the base of the covering (in particular its dimension) is rather limited, while the deck group is of the most importance. The results are also established for overdetermined elliptic systems, which in particular leads to Liouville theorems for polynomially growing holomorphic functions on abelian coverings of compact analytic manifolds. Analogous theorems hold for abelian coverings of compact combinatorial or quantum graphs.

math-ph↗
arXiv2004

A ground state alternative for singular Schrödinger operators

Let $\mathbf{a}$ be a quadratic form associated with a Schrödinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $Ω\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(Ω)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(Ω;\mathbb{R})$, or there is a sequence $ϕ_k\in C_0^{\infty}(Ω)$ and a function $ϕ>0$ satisfying $Lϕ=0$ such that $\mathbf{a}[ϕ_k]\to 0$, $ϕ_k\toϕ$ locally uniformly in $Ω\setminus\{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $Ω$. In the latter case, one has an inequality of Poincaré type: there exists $W>0$ such that for every $ψ\in C_0^\infty(Ω;\mathbb{R})$ satisfying $\int ψϕdx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u ψdx|^2$ for all $u\in C_0^\infty(Ω;\mathbb{R})$.

math.AP↗
arXiv2004

Existence of minimizers for Schrodinger operators under domain perturbations with application to Hardy's inequality

The paper studies the existence of minimizers for Rayleigh quotients $μ_Ω=\inf\frac{\int_Ω|\nabla u|^2}{\int_ΩV{|u|^2}} $, where $Ω$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative function that may have singularities on $\partialΩ$. As a model for our results one can take $Ω$ to be a Lipschitz cone and $V$ to be the Hardy potential $V(x)=\frac{1}{|x|^2} $.

math.AP↗
arXiv2002

Large time behavior of the heat kernel

In this paper we study the large time behavior of the (minimal) heat kernel $k_P^M(x,y,t)$ of a general time independent parabolic operator $L=u_t+P(x, \partial_x)$ which is defined on a noncompact manifold $M$. More precisely, we prove that $$\lim_{t\to\infty} e^{λ_0 t}k_P^{M}(x,y,t)$$ always exists. Here $λ_0$ is the generalized principal eigenvalue of the operator $P$ in $M$.

math.AP↗
arXiv2000

Integral representations and Liouville theorems for solutions of periodic elliptic equations

The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained.

math.AP↗