Local Boundedness, Maximum Principles, and Continuity of Solutions to Infinitely Degenerate Elliptic Equations with Rough Coefficients

Orlicz-Sobolev inequalities and the doubling condition

“…In R 2 consider the operator L defined as L = div A∇ with A = A(x, y) = diag{1, e 1/|x| }, (x, y) ∈ R 2 . Let d be a subunit metric associated to L (see, for example, [17,Definition 4] and [13,Chapter 7]), and µ = | | the Lebesgue measure. Then it was shown in [13,Conclusion 45] that the measure of a metric ball centered at the origin satisfies the following estimate…”

“…There has been a lot of interest in the theory of Sobolev-type inequalities in metric spaces in the past two decades, and it has seen great developments, see for instance [1,6,7,9,10,18] and references therein. A special interest in Sobolevtype inequalities arises in the study of regularity of solutions to certain classes of degenerate elliptic and parabolic PDEs, see for instance [1,, [2,4,11,13,14] and references therein. In the classical case of an elliptic operator, the Moser or DeGiorgi iteration technique can be used together with a Sobolev inequality to obtain higher regularity of solutions [16,3].…”

“…A more general, weaker, version of Sobolev inequality (an Orlicz-Sobolev inequality) has been recently proved for certain non-doubling metric measure spaces, and then successfully applied in the DeGiorgi iteration scheme to prove regularity of solutions to infinitely degenerate elliptic equations [13]. This poses a question of what types of Sobolev inequalities do imply the doubling condition.…”

“…More precisely, there exists a matrix Q, metric d, and a subset Ω ⊆ R n such that (5.3)w L Φ (µ B ) ≤ Cϕ(r (B)) [∇w] Q L 1 (µ B ) ,for all w ∈ W 1,1 Q (Ω, dx) with supp(w) ⊂ B, where (5.4) Φ(t) = t(ln t) α , α > 1, ∀t > expect the same lower bound on the superradius might be necessary, i.e. it should be possible to improve (5is a consequence of[13, Proposition 80]. More precisely, let n = 2, Q(x, y) = diag{1, f 2 (x)} where f (x) = exp(−1/|x| σ ), 0 < σ < 1, and let d be the metric subunit to Q, see [17, Definition 4].…”

Orlicz Sobolev Inequalities and the Doubling Condition

“…There has been a lot of interest in the theory of Sobolev-type inequalities in metric spaces in the past two decades, and it has seen great developments, see for instance [1,6,7,9,10,18] and references therein. A special interest in Sobolev-type inequalities arises in the study of regularity of solutions to certain classes of degenerate elliptic and parabolic PDEs, see for instance [1,, [2,4,11,13,14] and references therein. In the classical case of an elliptic operator, the Moser or DeGiorgi iteration technique can be used together with a Sobolev inequality to obtain higher regularity of solutions [16,3].…”