Use cases of Degenerate Elliptic Equations part4(Machine Learning 2024)
- Semilinear degenerate elliptic equation in the presence of singular nonlinearity(arXiv)
Author : Kaushik Bal, Sanjit Biswas
Abstract : Given Ω(⊆R1+m), a smooth bounded domain and a nonnegative measurable function f defined on Ω with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear degenerate elliptic equation with a singular nonlinearity given by:
−Δλu=fuν in Ωu>0 in Ωu=0 on ∂Ω
where the operator Δλ is given by
Δλu=uxx+|x|2λΔyu;(x,y)∈R×Rm
is known as the Grushin operator.
2. Harnack Inequalities and Continuity of Solutions to Infinitely Degenerate Elliptic Equations(arXiv)
Author : Lyudmila Korobenko, Cristian Rios, Eric Sawyer, Ruipeng Shen
Abstract : Using Moser iteration in subunit metric spaces we prove a weak version of Harnack’s inequality for weak subsolutions to certain infinitely degenerate elliptic divergence form equations. This weak Harnack inequality is then used to prove continuity of weak solutions under some additional restriction on the degeneracy. For example, we consider the family {fk,σ}k∈N,σ>0 of functions on the real line,
fk,σ(x)=|x|(ln(k)1|x|)σ, −∞<x<∞,
that are infinitely degenerate at the origin, and derive conditions on the parameters k and σ under which all weak solutions to the associated infinitely degenerate planar quasilinear equations of the form
divA(x,y,u)gradu=φ(x,y), A(x,y,z)∼[100fk,σ(x)2],
with rough data A and φ, are continuous, e.g. k=3 and 0<σ<1
