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

@article{Grunau2021OptimalEF,
  title={Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients},
  author={Hans-Christoph Grunau},
  journal={Archiv der Mathematik},
  year={2021}
}
  • H. Grunau
  • Published 2021
  • Mathematics
  • Archiv der Mathematik
Estimates 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… Expand
2 Citations
An operator theoretic approach to uniform (anti-)maximum principles
Maximum principles and uniform anti-maximum principles are a ubiquitous topic in PDE theory that is closely tied to the Krein–Rutman theorem and kernel estimates for resolvents. We take up aExpand
Positivity for the clamped plate equation under high tension
In this article we consider positivity issues for the clamped plate equation with high tension γ > 0. This equation is given by ∆2u − γ∆u = f under clamped boundary conditions. Here we show, thatExpand

References

SHOWING 1-10 OF 19 REFERENCES
Optimal estimates from below for biharmonic Green functions
Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary C 4,γ-smooth domains. Maximum principles do not exist for fourth orderExpand
Dominance of positivity of the Green's function associated to a perturbed polyharmonic dirichlet boundary value problem by pointwise estimates
In this work we study the behaviour of the Green function for a linear higher-order elliptic problem. More precisely, we consider the Dirichlet boundary value problem in a bounded C2m,γ-smooth domainExpand
Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators
We study fundamental solutions of elliptic operators of order $$2m\ge 4$$ 2 m ≥ 4 with constant coefficients in large dimensions $$n\ge 2m$$ n ≥ 2 m , where their singularities become unbounded. ForExpand
Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains
Summary The main result in this paper is that the solution operator for the bi-Laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive partExpand
Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor aExpand
Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions
Higher order elliptic partial dierential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shownExpand
Classical solutions up to the boundary to some higher order semilinear Dirichlet problems
Abstract We consider the semilinear Dirichlet problem ( − Δ ) m u + g ( ⋅ , u ) = f in bounded domains Ω ⊂ R n under homogeneous Dirichlet boundary conditions ( ∂ ∂ ν ) i u = 0 for i = 0 , … , m − 1Expand
Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems
Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence one obtains optimal weighted L p -L q -regularityExpand
ISOLATION OF SINGULARITIES OF THE GREEN'S FUNCTION
A method is given for constructing the Green's function for general boundary-value problems for elliptic equations. Pointwise estimates are proved for the Green's function in a closed region. AExpand
A biharmonic converse to Krein–Rutman: a maximum principle near a positive eigenfunction
The Green function $$G_0(x,y)$$ G 0 ( x , y ) for the biharmonic Dirichlet problem on a smooth domain $$\Omega $$ Ω , that is $$\Delta ^{2}u=f$$ Δ 2 u = f in $$\Omega $$ Ω with $$ u=u_{n}=0 $$ u = uExpand
...
1
2
...