Monotonicity of non-Liouville property for positive solutions of skew product elliptic equations

2019 ◽  
Vol 150 (3) ◽  
pp. 1429-1449
Author(s):  
Minoru Murata ◽  
Tetsuo Tsuchida
Keyword(s):  
Elliptic Operator ◽  
Martin Boundary ◽  
Monotonicity Property ◽  
Skew Product ◽  
Liouville Property ◽  
Subcritical Case ◽  
Minimal Growth ◽  
Potential Term ◽  
Order Elliptic Operator ◽  
Martin Compactification

AbstractWe consider a second-order elliptic operator L in skew product of an ordinary differential operator L1 on an interval (a, b) and an elliptic operator on a domain D2 of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality of L in terms of a positive solution having minimal growth at η (η = a, b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel for L on the basis of [24]; in particular, the Martin boundary over η is either one point or a compactification of D2, which depends on whether an associated integral near η diverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary over η does not become smaller as the potential term of L1 becomes larger near η.

Partial avaraging of the nondivergence second order elliptic operator

2016 ◽  
Vol 31 (3) ◽  
pp. 47-53
Author(s):  
M.M. Sirazhudinov ◽  
◽  
S.P. Dzhamaludinova ◽  
M.E. Mahmudova ◽  
◽  
...  
Keyword(s):  
Elliptic Operator ◽  
Second Order ◽  
Order Elliptic Operator

scholarly journals Green’s Function Estimates for Time-Fractional Evolution Equations

2019 ◽  
Vol 3 (2) ◽  
pp. 36
Author(s):  
Ifan Johnston ◽  
Vassili Kolokoltsov
Keyword(s):  
Green’S Function ◽  
Elliptic Operator ◽  
Green's Function ◽  
Green's Functions ◽  
Evolution Equations ◽  
Green’S Functions ◽  
Variable Coefficients ◽  
Second Order ◽  
Fractional Evolution Equations ◽  
Order Elliptic Operator

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.

scholarly journals Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

2016 ◽  
Vol 28 (5) ◽  
pp. 823-856 ◽  
Author(s):  
Jun Cao ◽  
Svitlana Mayboroda ◽  
Dachun Yang
Keyword(s):  
Elliptic Operator ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Divergence Form ◽  
Higher Order ◽  
Square Functions ◽  
Maximal Interval ◽  
Measurable Coefficients ◽  
Order Elliptic Operator ◽  
Higher Order Elliptic Operator

AbstractLet L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and ${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents ${q\in[1,\infty]}$ such that the semigroup ${\{e^{-tL}\}_{t>0}}$ is bounded on ${L^{q}(\mathbb{R}^{n})}$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces ${H_{L}^{p}(\mathbb{R}^{n})}$ for all ${p\in(0,p_{+}(L))}$, which when ${p=1}$, answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L.

scholarly journals Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

2016 ◽  
Vol 146 (6) ◽  
pp. 1115-1158 ◽  
Author(s):  
Denis Borisov ◽  
Giuseppe Cardone ◽  
Tiziana Durante
Keyword(s):  
Boundary Conditions ◽  
Elliptic Operator ◽  
Rate Of Convergence ◽  
Elliptic Operators ◽  
Resolvent Convergence ◽  
Order Elliptic Operator ◽  
Classical Boundary ◽  
Norm Resolvent Convergence ◽  
Operator Norms ◽  
Infinite Planar

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

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