arXiv subjects

Igor E. Verbitsky

Publications and source records attributed to Igor E. Verbitsky.

At least 19 recordsLinked to original sources

arXiv2022

Uniqueness of entire solutions to quasilinear equations of p-Laplace type

We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} where $σ$ is a nonnegative locally finite measure in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and ${\rm div}\, \mathcal{A}(x,\nabla u)$ is the $\mathcal{A}$-Laplace operator, under standard growth and monotonicity assumptions of order $p$ ($1<p<\infty$) on $\mathcal{A}(x, ξ)$ ($x, ξ\in \mathbb{R}^n$); the model case $\mathcal{A}(x, ξ)=ξ| ξ|^{p-2}$ corresponds to the $p$-Laplace operator $Δ_p$ on $\mathbb{R}^n$. Our main results establish uniqueness of solutions to a similar problem, \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = σu^q +μ, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\ \displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} in the sub-natural growth case $0<q<p-1$, where $μ, σ$ are nonnegative locally finite measures in $\mathbb{R}^n$, absolutely continuous with respect to the $p$-capacity, and $\mathcal{A}(x, ξ)$ satisfies an additional homogeneity condition, which holds in particular for the $p$-Laplace operator.

math.AP↗

arXiv2022

Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0 < q < 1$, $σ\ge 0$ is a Radon measure in $Ω$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $Ω\timesΩ$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{\rm loc} (Ω, σ)$. Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c. \] Here $0<q<1$, $μ, σ\ge 0$ are Radon measures, and $Ω$ is a bounded uniform domain in ${\mathbb R}^n$, if $0 < α\le 2$, or the entire space ${\mathbb R}^n$, a ball or half-space, if $0 < α<n$. Analogues of these results are presented for elliptic equations involving the $p$-Laplace operator on the entire space ${\mathbb R}^n$, \[ -Δ_p u = σu^q + μ\quad \textrm{in} \,\, {\mathbb R}^n, \qquad \liminf_{x\to \infty} u(x)=0, \] where $0<q<p-1$, and $μ, σ\ge 0$ are Radon measures. More general quasilinear equations with $\mathcal{A}$-Laplace operators ${\rm div} \mathcal{A}(x, \nabla u)$ in place of $Δ_p$ are covered as well.

math.AP↗

arXiv2022

Global pointwise estimates of positive solutions to sublinear equations

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω,\] for $0 < q < 1$, where $σ\ge 0$ is a measurable function, or a Radon measure, $f \ge 0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel $G$ on $Ω\timesΩ$. Our main results, which include the existence criteria and uniqueness of solutions, hold for quasi-metric, or quasi-metrically modifiable kernels $G$. As a consequence, we obtain bilateral estimates, along with the existence and uniqueness, for positive solutions $u$, possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, \[ (-Δ)^{\fracα{2}} u = σu^q + μ\quad \textrm{in} \,\, Ω, \qquad u=0 \, \, \textrm{in} \,\, Ω^c, \] where $0<q<1$, and $μ, σ\ge 0$ are measurable functions, or Radon measures, on a bounded uniform domain $Ω\subset \mathbf{R}^n$ for $0 < α\le 2$, or on the entire space $\mathbf{R}^n$, a ball or half-space, for $0 < α<n$.

math.AP↗

arXiv2021

BMO solutions to quasilinear equations of $p$-Laplace type

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-Δ_{p} u = μ$ in $\mathbb{R}^n$, $u\ge 0$, where $μ$ is a locally finite Radon measure, and $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). We also characterize BMO solutions to equations $-Δ_{p} u = σu^{q} + μ$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $μ$ and $σ$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in place of $Δ_{p}$.

math.AP↗

arXiv2020

Positive solutions and harmonic measure for Schrödinger operators in uniform domains

We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = ωu \, \,& & \mbox{in} \, \, Ω, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial Ω, \end{aligned} \right. \end{equation*} in a bounded uniform domain $Ω\subset {\bf R}^n$, where $ω$ is a locally finite Borel measure in $Ω$, and $f\ge 0$ is integrable with respect to harmonic measure $d H^{x}$ on $\partialΩ$. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of $M^{*} (m ω)(z)=\int_ΩM(x, z) m(x)\, d ω(x)$ on $\partialΩ$ with respect to $f \, d H^{x_0}$, where $M(x, \cdot)$ is Martin's function with pole at $x_0\in Ω, m(x)=\min (1, G(x, x_0))$, and $G$ is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schrödinger operator $-\triangle - ω$ on $Ω$, and in the case $f=1$, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.

math.AP↗

arXiv2019

Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

We discuss recent advances in the theory of quasilinear equations of the type $ -Δ_{p} u = σu^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $σ$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u \in \text{BMO}(\mathbb{R}^n)$, $u \in L^r_{\rm loc}(\mathbb{R}^n)$, etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.

math.AP↗

arXiv2018

Solutions to sublinear elliptic equations with finite generalized energy

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = σu^{q} + μ\quad \text{in} \;\; Ω, \] in the sublinear case $0 0$. In this case $u \in L^{γ+q}(Ω, σ)\cap L^γ(Ω, μ)$, where $γ=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $σ$, $μ$ are nonnegative functions (or Radon measures), on an arbitrary domain $Ω\subseteq \mathbb{R}^n$ which possesses a positive Green function associated with $\mathcal{L}$. When $0<γ\leq 1$, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space $\dot{W}_{0}^{1,p}(Ω)$ for $1<p\leq 2$.

math.AP↗

arXiv2018

Wolff's inequality for intrinsic nonlinear potentials and quasilinear elliptic equations

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth case $0<q< p-1$, where $Δ_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian, and $σ$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. As an application, we give a necessary and sufficient condition for the existence of a positive solution $u \in L^{r}(\mathbb{R}^{n})$ ($0<r<\infty$) to this problem, which was open even in the case $p=2$. Our version of Wolff's inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood-Paley spaces $f^{p, q}(σ)$ defined in terms of characteristic functions of dyadic cubes in $\mathbb{R}^n$.

math.AP↗

arXiv2018

Solutions in Lebesgue spaces to nonlinear elliptic equations with sub-natural growth terms

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q< p-1$, where $Δ_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $σ$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the $\mathcal{A}$-Laplacian $\text{div} \mathcal{A}(x,\nabla u)$, and the fractional Laplacian $(-Δ)^α$ on $\mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\text{div}(\mathcal{A} \nabla u)$ on an arbitrary domain $Ω\subseteq \mathbb{R}^n$ with a positive Green function.

math.AP↗

arXiv2018

On two-weight norm inequalities for positive dyadic operators

Let $σ$ and $ω$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(fσ) \rVert_{L^q(ω)}\leq C \lVert f \rVert_{L^p(σ)}, \quad \text{for all} \, \, f \in L^p(σ), $$ for both the positive summation operators $T=T_λ(\cdot σ)$ and positive maximal operators $T=M_λ(\cdot σ)$. Here, for a family $\{λ_Q\}$ of non-negative reals indexed by the dyadic cubes $Q$, these operators are defined by $$ T_λ(fσ):=\sum_Q λ_Q \langle f\rangle^σ_Q 1_Q \quad\text{ and } \quad M_λ(fσ):=\sup_Q λ_Q \langle f\rangle^σ_Q 1_Q, $$ where $\langle f\rangle^σ_Q:=\frac{1}{σ(Q)} \int_Q |f| d σ.$ We obtain new characterizations of the two-weight norm inequalities in the following cases: 1. For $T=T_λ(\cdotσ)$ in the subrange $q<p$. Under the additional assumption that $σ$ satisfies the $A_\infty$ condition with respect to $ω$, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. 2. For $T=M_λ(\cdot σ)$ in the subrange $q<p$. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. 3. For the summation operators $T=T_λ(\cdotσ)$ in the subrange $1<q<p$. We characterize the inequality for summation operators by means of related inequalities for maximal operators $T=M_λ(\cdot σ)$. This maximal-type characterization is an alternative to the known potential-type characterization.

math.CA↗

arXiv2018

Existence of the gauge for fractional Laplacian Schrödinger operators

Let $Ω\subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $ω$ is a locally finite Borel measure on $Ω$. For $α\in (0,2)$, define the fractional Laplacian $(-\triangle )^{α/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $α$ on $Ω$. Define $T(u) = G(u ω).$ If $\Vert T \Vert_{L^2(ω) \rightarrow L^2 (ω)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{α/2, 2}_0 (Ω)$ of \[ (-\triangle)^{α/2} u = u ω+ ν\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, Ω^c, \] for $ν$ in the dual Sobolev space $L^{-α/2, 2} (Ω)$. If $Ω$ is a bounded $C^{1,1}$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $ν= χ_Ω$. These estimates are used to study the existence of a solution $u_1$ (called the "gauge") of the integral equation $u_1=1+G(u_1 ω)$ corresponding to the problem \[ (-\triangle)^{α/2} u = u ω\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u \geq 0 \,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=1 \,\,\, \mbox{on} \,\,\, Ω^c . \] We show that if $\Vert T \Vert <1$, then $u_1$ always exists if $0<α<1$. For $1 \leq α<2$, a solution exists if the norm of $T$ is sufficiently small. We also show that the condition $\Vert T \Vert <1$ does not imply the existence of a solution if $1 < α<2$.

math.AP↗

arXiv2018

A sublinear version of Schur's lemma and elliptic PDE

We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}ν\Vert_{L^q(Ω, dσ)} \le C \Vert ν\Vert, \quad \text{ for all } ν\in \mathcal{M}^+(Ω), \] along with its weak-type analogue, for $0 < q < 1$, where $\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{M}^+(Ω)$ denotes the class of positive Radon measures in $Ω$; $σ, ν\in \mathcal{M}^+(Ω)$, and $||ν||=ν(Ω)$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $σ$ for which such an embedding holds. The strong-type $(1,q)$-inequality for $0<q<1$ is closely connected with existence of a positive function $u$ such that $u \ge \mathbf{G}(u^q σ)$, i.e., a supersolution to the integral equation \[ u - \mathbf{G}(u^q σ) = 0, \quad u \in L^q_{\rm loc} (Ω, σ). \] This study is motivated by solving sublinear equations involving the fractional Laplacian, \[ (-Δ)^{\fracα{2}} u - u^q σ= 0\] in domains $Ω\subseteq \mathbf{R}^n$ which have a positive Green function $G$, for $0 < α< n$.

math.AP↗

arXiv2017

Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} + μ\quad \text{on} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where $Δ_{p}$ ($1<p<\infty$) is the $p$-Laplacian, and $σ$, $μ$ are positive Borel measures on $\mathbb{R}^n$. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for the fractional Laplace operator $(-Δ)^α$ in place of $-Δ_{p}$, on $\mathbb{R}^n$ for $0<α<\frac{n}{2}$, and on an arbitrary domain $Ω\subset \mathbb{R}^n$ with positive Green's function in the classical case $α= 1$.

math.AP↗

arXiv2017

Sublinear equations and Schur's test for integral operators

We study weighted norm inequalities of $(p,r)$-type, $ \Vert \mathbf{G} (f \, d σ) \Vert_{L^r(Ω, dσ)} \le C \Vert f \Vert_{L^p(Ω, σ)}, \quad \forall \, f \in L^p(σ),$ for $0 < r < p$ and $p>1$, where $\mathbf{G}(f d σ)(x)=\int_ΩG(x, y) f(y) d σ(y)$ is an integral operator associated with a nonnegative kernel $G$ on $Ω\times Ω$, and $σ$ is a locally finite positive measure in $Ω$. We show that this embedding holds if and only if $\int_Ω(\mathbf{G} σ)^{\frac{pr}{p-r}} d σ<+\infty,$ provided $G$ is a quasi-symmetric kernel which satisfies the weak maximum principle. In the case $p=\frac{r}{q}$, where $0 q$, to the the sublinear integral equation $ u - \mathbf{G}(u^q \, d σ) = 0, \quad u \ge 0.$ We also give some counterexamples in the end-point case $p=1$, which corresponds to solutions $u \in L^q (Ω, σ)$ of this integral equation. These problems appear in the investigation of weak solutions to the sublinear equation involving the (fractional) Laplacian, $(-Δ)^α u - σ\, u^q = 0, \quad u \ge 0,$ for $0<q<1$ and $0 < α< \frac{n}{2}$ in domains $Ω\subseteq \mathbb{R}^n$ with a positive Green function.

math.AP↗

arXiv2017

Two-weight $L^p\to L^q$ bounds for positive dyadic operators in the case $0<q< 1 \le p<\infty$

Let $σ$, $ω$ be measures on $\mathbb{R}^d$, and let $\{λ_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize the two-weight norm inequality, \begin{equation*} \lVert T_λ(fσ)\rVert_{L^q(ω)}\le C \, \lVert f \rVert_{L^p(σ)}\quad \text{for every $f\in L^p(σ)$,} \end{equation*} for the positive dyadic operator \begin{equation*} T_λ(fσ):= \sum_{Q\in \mathcal{D}} λ_Q \, \Big(\frac{1}{σ(Q)} \int_Q f\mathrm{d}σ\Big) \, 1_Q \end{equation*} in the difficult range $0<q<1 \le p<\infty$ of integrability exponents. This range of the exponents $p, q$ appeared recently in applications to nonlinear PDE, which was one of the motivations for our study. Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials $I_α(f σ) = (-Δ)^{-\fracα{2}} (fσ) $ ($0<α<d$), since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of $p, q$ has been characterized previously only in the special case where $σ$ is Lebesgue measure.

math.CA↗

arXiv2016

Weighted norm inequalities of (1,q)-type for integral and fractional maximal operators

We study weighted norm inequalities of $(1,q)$- type for $0<q<1$, $\Vert \mathbf{G} ν\Vert_{L^q(Ω, d σ)} \le C \, \Vert ν\Vert, \quad \text{for all positive measures $ν$ in $Ω$},$ along with their weak-type counterparts, where $\Vert ν\Vert=ν(Ω)$, and $G$ is an integral operator with nonnegative kernel, $\mathbf{G} ν(x) = \int_ΩG(x, y) d ν(y).$ These problems are motivated by sublinear elliptic equations in a domain $Ω\subset\mathbb{R}^n$ with non-trivial Green's function $G(x, y)$ associated with the Laplacian, fractional Laplacian, or more general elliptic operator. We also treat fractional maximal operators $M_α$ ($0\le α<n$) on $\mathbb{R}^n$, and characterize strong- and weak-type $(1,q)$-inequalities for $M_α$ and more general maximal operators, as well as $(1,q)$-Carleson measure inequalities for Poisson integrals.

math.AP↗

arXiv2016

Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations

We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian $(-Δ)^α$ for $0<α<\frac{n}{2}$ are treated as well, including explicit estimates for radially symmetric $σ$. Our results are new even in the classical case $p=2$ and $α=1$.

math.AP↗

arXiv2015

Fourier multipliers for weighted $L^{2}$ spaces with Lévy-Khinchin-Schoenberg weights

We present a class of weight functions $ w$ on the circle $ \mathbb{T}$, called Lévy-Khinchin-Schoenberg (LKS) weights, for which we are able to completely characterize (in terms of a capacitary inequality) all Fourier multipliers for the weighted space $ L^{2}(\mathbb{T},w)$. We show that the multiplier algebra is nontrivial if and only if $ 1/w\in L^{1}(\mathbb{T})$, and in this case multipliers satisfy the Spectral Localization Property (no "hidden spectrum"). On the other hand, the Muckenhoupt $ (A_{2})$ condition responsible for the basis property of exponentials $ (e^{ikx})$ is more or less independent of the Spectral Localization Property and LKS requirements. Some more complicated compositions of LKS weights are considered as well.

math.FA↗