Estimates in Morrey Spaces and Hölder Continuity for Weak Solutions to Degenerate Elliptic Systems

Abstract

In this paper, we establish gradient estimates in Morrey spaces and Hölder continuity for weak solutions of the following degenerate elliptic systemwhere X 1 , · · · , X q are real smooth vector fields satisfying Hörmander's condition, coefficients a αβ i j ∈ V M O X ∩ L ∞ ( ), α, β = 1, 2, · · · , q, i, j = 1, 2, · · · , N , X * α is the transposed vector field of X α .

Hölder regularity for weak solutions of diagonal divergence quasilinear degenerate elliptic systems

“…2 By Theorem 3.1 and Theorem 2.2, we easily know that the following result is true by using a similar proof as in [5,Theorem 3.2]. …”

“…Di Fazio and Fanciullo in [4] showed gradient estimates in Morrey spaces when p = 2 and Hölder regularity by using an isomorphic relationship for a divergence linear degenerate elliptic system in Lipschitz vector fields. The interior Hölder regularity for weak solutions of linear degenerate elliptic systems was concluded by Dong and Niu in [5] with energy estimates and reverse Hölder inequality on the homogeneous spaces.…”

Regularity for weak solutions to nondiagonal quasilinear degenerate elliptic systems

“…By [26], we know that assertions hold. Proof: Since u 0 (y) = v(Sy) satisfies −X * α X β u j 0 = 0 in B R , we have by [9] that…”

“…Gao, Niu and Wang in [11] settled partial Hölder regularity for weak solutions to degenerate quasilinear elliptic systems with the coefficients belonging to V MO ∩ L ∞ and the low term satisfying the natural condition. We mention that those systems in [8,9,11,26] are all diagonal.…”

Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

“…Based on Hömander's fundamental work [11], there has been tremendous work on degenerate PDEs arising from non-commuting vector fields; see, for example, [12][13][14][15][16][17][18][19][20][21]. Di Fazio and Fanciullo in [14] obtained gradient estimates for weak solutions to linear diagonal elliptic systems with bounded VMO coefficients.…”