scholarly journals Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

2020 ◽  
Vol 1 (1) ◽  
pp. 52-65
Author(s):  
Rabil Ayazoglu- Sidika S¸ ule S¸ ener · Tuba Agırman Aydın
Keyword(s):  
Existence Of Solutions ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Resonant Problem ◽  
General Elliptic

scholarly journals Nontangential Estimates on Layer Potentials and the Neumann Problem for Higher-Order Elliptic Equations

2020 ◽  
Author(s):  
Ariel Barton ◽  
Steve Hofmann ◽  
Svitlana Mayboroda
Keyword(s):  
Neumann Problem ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Layer Potentials ◽  
The Neumann Problem

Abstract We solve the Neumann problem, with nontangential estimates, for higher-order divergence form elliptic operators with variable $t$-independent coefficients. Our results are accompanied by nontangential estimates on higher-order layer potentials.

Pointwise lower bounds on the heat kernels of higher order elliptic operators

1999 ◽  
Vol 125 (1) ◽  
pp. 105-111 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Space Dimension ◽  
Adjoint Operator ◽  
Integral Kernel ◽  
Divergence Form ◽  
Higher Order ◽  
Elliptic Operators ◽  
Heat Kernels ◽  
Second Order Elliptic Operators

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

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