scholarly journals Sharp estimates for iterated Green functions

2002 ◽  
Vol 132 (1) ◽  
pp. 91-120 ◽  
Author(s):  
H.-Ch. Grunau ◽  
G. Sweers
Keyword(s):  
Dirichlet Boundary ◽  
Elliptic Systems ◽  
Elliptic Operators ◽  
Dirichlet Boundary Conditions ◽  
Green Functions ◽  
Pointwise Estimates ◽  
Polyharmonic Operator ◽  
Sharp Estimates ◽  
Conditioned Brownian Motion ◽  
Second Order Elliptic Operators

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second-order elliptic operators, these estimates hold true on bounded C1,1 domains. For higher-order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to non-cooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

scholarly journals Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

2021 ◽  
Author(s):  
Hans-Christoph Grunau
Keyword(s):  
Principal Part ◽  
Present Note ◽  
Variable Coefficients ◽  
Higher Order ◽  
Elliptic Operators ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Green Functions ◽  
Polyharmonic Operator ◽  
Bounded Smooth

AbstractEstimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order $$2m\ge 4$$ 2 m ≥ 4 have been shown so far only when the principal part of L is the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m . In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.

Estimates on the Green's function of second-order elliptic operators in ℝN

1998 ◽  
Vol 128 (5) ◽  
pp. 1033-1051
Author(s):  
Adrian T. Hill
Keyword(s):  
Green’S Function ◽  
Heat Kernel ◽  
Hypergeometric Function ◽  
Green's Function ◽  
Confluent Hypergeometric Function ◽  
Elliptic Operators ◽  
Polar Coordinates ◽  
Pointwise Estimates ◽  
Image Position ◽  
Second Order Elliptic Operators

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equationfor λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

scholarly journals Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems withp-Laplacian with Nonlocal Sources

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
Zhoujin Cui ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Local Theory ◽  
Smooth Bounded Domain ◽  
Nonlocal Sources ◽  
Nonexistence Result ◽  
Evolution Systems

This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

Phragmen–Lindelöf theorems and the asymptotic behaviour of solutions of quasilinear elliptic equations in slabs

2000 ◽  
Vol 130 (2) ◽  
pp. 335-373 ◽  
Author(s):  
Z. Jin ◽  
K. Lancaster
Keyword(s):  
Asymptotic Behaviour ◽  
Boundary Data ◽  
Dirichlet Boundary ◽  
Elliptic Operators ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Elliptic Equations ◽  
Compact Set ◽  
Asymptotic Behaviour Of Solutions ◽  
Quasilinear Elliptic ◽  
Dirichlet Boundary Data

The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n > 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.

scholarly journals AN ANALYTICAL APPROACH TO HEAT KERNEL ESTIMATES ON STRONGLY RECURRENT METRIC SPACES

2008 ◽  
Vol 51 (1) ◽  
pp. 171-199 ◽  
Author(s):  
Jiaxin Hu
Keyword(s):  
Dirichlet Boundary ◽  
Metric Spaces ◽  
Dirichlet Forms ◽  
Upper Bounds ◽  
Heat Kernels ◽  
Dirichlet Boundary Conditions ◽  
Green Functions ◽  
Compact Embedding ◽  
Kernel Estimates ◽  
Compact Metric Spaces

AbstractIn this paper we prove that sub-Gaussian estimates of heat kernels of regular Dirichlet forms are equivalent to the regularity of measures, two-sided bounds of effective resistances and the locality of semigroups, on strongly recurrent compact metric spaces. Upper bounds of effective resistances imply the compact embedding theorem for domains of Dirichlet forms, and give rise to the existence of Green functions with zero Dirichlet boundary conditions. Green functions play an important role in our analysis. Our emphasis in this paper is on the analytic aspects of deriving two-sided sub-Gaussian bounds of heat kernels. We also give the probabilistic interpretation for each of the main analytic steps.

On the existence of multiple positive solutions to some superlinear systems

2012 ◽  
Vol 142 (1) ◽  
pp. 39-59 ◽  
Author(s):  
M. Chhetri ◽  
S. Raynor ◽  
S. Robinson
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Convex Shape ◽  
Smooth Bounded Domain ◽  
Image Position

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the formon a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

scholarly journals The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

2019 ◽  
Vol 19 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Sergio Fernández-Rincón ◽  
Julián López-Gómez
Keyword(s):  
Singular Perturbation ◽  
Boundary Conditions ◽  
Distance Function ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Logistic Equations ◽  
Perturbation Problem ◽  
Second Order Elliptic Operators

Abstract This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, {d\mathcal{L}u=uh(u,x)} , under non-classical mixed boundary conditions, {\mathcal{B}u=0} on {\partial\Omega} . Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of {\partial\Omega} through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

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