# Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators

@article{Grunau2019DifferencesBF, title={Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators}, author={Hans-Christoph Grunau and Giulio Romani and Guido Sweers}, journal={Mathematische Annalen}, year={2019}, pages={1-54} }

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. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as $$n\ge 3$$ n ≥ 3 , the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m may no longer serve as a prototype for the general elliptic operator… Expand

#### 6 Citations

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

- Mathematics
- 2021

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… Expand

A biharmonic converse to Krein–Rutman: a maximum principle near a positive eigenfunction

- Mathematics
- Positivity
- 2019

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 = u… Expand

Classical solutions up to the boundary to some higher order semilinear Dirichlet problems

- Mathematics
- 2021

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 − 1… Expand

A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

- Mathematics
- 2020

It is well known that for higher order elliptic equations the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that… Expand

Note on a sign-dependent regularity for the polyharmonic Dirichlet problem

- Mathematics
- 2020

A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic… Expand

Positivity for the clamped plate equation under high tension

- Mathematics
- 2021

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, that… Expand

#### References

SHOWING 1-10 OF 22 REFERENCES

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

- Mathematics
- 2007

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 a… Expand

Elliptic Partial Differential Equations of Second Order

- Physics
- 1997

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations… Expand

Dominance of positivity of the Green's function associated to a perturbed polyharmonic dirichlet boundary value problem by pointwise estimates

- Mathematics
- 2015

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 domain… Expand

Optimal estimates from below for biharmonic Green functions

- Mathematics
- 2010

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 order… Expand

The analysis of linear partial differential operators

- Mathematics
- 1990

the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certainproducts. Many products that you buy can be… Expand

Plane Waves and Spherical Means: Applied To Partial Differential Equations

- Mathematics
- 1981

The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have… Expand

An Introduction to Harmonic Analysis

- Mathematics
- 1968

1. Fourier series on T 2. The convergence of Fourier series 3. The conjugate function 4. Interpolation of linear operators 5. Lacunary series and quasi-analytic classes 6. Fourier transforms on the… Expand

Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains

- Mathematics
- 2010

Models of Higher Order.- Linear Problems.- Eigenvalue Problems.- Kernel Estimates.- Positivity and Lower Order Perturbations.- Dominance of Positivity in Linear Equations.- Semilinear Problems.-… Expand

Introductory Functional Analysis With Applications

- Mathematics
- 1978

Metric Spaces. Normed Spaces Banach Spaces. Inner Product Spaces Hilbert Spaces. Fundamental Theorems for Normed and Banach Spaces. Further Applications: Banach Fixed Point Theorem. Spectral Theory… Expand