Higher Integrability of the Gradient in Degenerate Elliptic Equations

Abstract

We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta-Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution.

Radiative effects for some bidimensional thermoelectric problems

“…The existence and uniqueness of φ and Θ are ensured by Using the estimates (56) and (17) in accordance with Theorem 2.2, and taking R ≥ g p,Γ N , we deduce…”

Subordinated Gaussian Random Fields in Elliptic Partial Differential Equations

“…for any k reg ≥ 1 and a deterministic, positive constant ϑ > 0. We note that although [19,Theorem 1] suggests a dependence of the parameter ϑ = ϑ(D, f L q (D) , a − , A) on the constant A, this dependence is not numerically detectable for our diffusion coefficient, as numerical experiments show. Of course, it depends on the other parameters D, f L q (D) and a − .…”

“…Φ 1 ≡ 0) and a is constant. Another important result is given in [19]. It follows by [19,Theorem 1] that under the assumption that there exists q > 2 with f ∈ L q (D) P − a.s. there exists a constant C = C(D, f L q (D) , a − , A) and a positive number ϑ = ϑ(D, f L q (D) , a − , A) > 0 only depending on the indicated parameters, such that:…”

Results for a turbulent system with unbounded viscosities: Weak formulations, existence of solutions, boundedness and smoothness

“…Assume now that the condition (3) in Corollary 1 is fulfilled. By using (38) together with the Sobolev Injection Theorem we get k ∈ L 6 (Ω) and thus ν(k) ∈ L 1 (Ω). Then we can conclude the proof of Corollary 1 by using Proposition 9 in the Appendix I: (u, k) is a distributional solution of (P).…”

“…Moreover W k is not necessarily complete and a function in H k does not always have a uniquely defined gradient (see [14]). If we assume that ν(k) ∈ L 1 (Ω) then W k is complete and in fact H k ⊂ W k are Hilbert spaces (see [6,14,13]) when they are equipped with the scalar product…”