Solutions to degenerate elliptic equations: existence, boundedness, regularity
A talk in the BI.discrete series by
David Cruz-Uribe from University of Alabama
| Abstract: | It is a classical result, due to Trudinger, Nash, Moser, de Georgi, and
others, that if Q is a uniformly elliptic matrix, and f ∈ L2(Ω), then there exists a
weak solution u of the Dirichlet problem,
−Div (Q∇u) = f for x ∈ Ω,
u = 0 for x ∈ ∂Ω.
If we further assume that f ∈ Lq(Ω), q > n
2 , then solutions are bounded functions
and satisfy
∥u∥L∞(Ω) ≤ C∥f∥Lq(Ω).
This result is sharp in the sense that if q = n
2 , then there exists f ∈ L
n
2 (Ω) such that
this inequality fails even for the Laplacian (Q = I). Finally, the solutions are locally
H¨older continuous: given a ball B such that 2B ⊂ Ω, there exists 0 < α < 1 such
that u ∈ Cα(B). Corresponding results hold if we consider the differential equation
with lower order terms.
In this talk we will discuss results from a large project to systematically extend
this theory to the degenerate elliptic equation
−v−1 Div (Q∇u) = f for x ∈ Ω,
u = 0 for x ∈ ∂Ω,
where Q is no longer uniformly elliptic but satisfies the degenerate ellipticity condition
w(x)|ξ|2 ≤ ⟨Qξ, ξ⟩ ≤ v(x)|ξ|2, ξ ∈ Rn.
We will discuss existence, uniqueness, and boundedness of solutions to this equation
and the corresponding equation with lower order terms. A great deal of work has
gone into determining the minimal hypotheses required to establish these results.
Central has been the existence of a global degenerate Sobolev inequality
Z
Ω
|φ|σpv dx
1
σp
≤
Z
Ω
|
p
Q∇φ|p dx
1
p
,
where σ ≥ 1 and φ is a smooth function of compact support. We will discuss very
recent work using a generalization of Rubio de Francia extrapolation to prove such
inequalities with minimal assumptions on Q and v.
This research is in collaboration with Scott Rodney, Cape Breton University, Sydney
Canada, and Yusuf Zeren and his students S¸eyma C¸ etin and Feyza Elif Dal at
Yıldız Technical University, Istanbul T¨urkiye. Within the CRC this talk is associated to the project(s): A7 |