Higher-Order Regularity for the Solutions of Some Degenerate Quasilinear Elliptic Equations in the Plane

Abstract

Abstract. Local C/, and W 2+," (k _> 1,/ > 0, and v >_ 1) regularity is established for the solutions of a class of degenerate quasilinear elliptic equations, which include the p-Laplacian. Unlike the known local regularity results for such equations, k is larger than 2 in many notable cases. These results generalize those in [13], which were established only for the p-Laplacian. Furthermore, local results are extended to obtain a global regularity result in some cases. Global results of this type are essential in proving optimal error bounds for the finite element approximation of such equations.

Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems

“…For instance, for sufficiently regular data, global C 1,α regularity is established in [19] for the p-Laplacian. Higher order regularity (like H 2 ( )) is investigated in [21]- [23]. In the rest of the paper, we assume that the solution u is continuous.…”

“…However, the regularity u ∈ W 3,1 ( ) ∩ C 2,1+2/p (¯ ) was only shown for the p-harmonic functions where f = 0, see [21]. Indeed it does not seem that such higher regularity is in general achievable for the p-Laplacian even with very smooth data.…”

“…During the last decade, there has been significant progress in finite element approximation of this class of degenerate equations, see [1], [2], [10], [6]- [5], [9], [15], [20], and [21]- [31]. In particular, extensive research has been carried out for the finite element approximation of the p-Laplacian, since it is believed that this simpler equation contains most of the essential difficulties in the studies of finite element approximation of this class of degenerate systems.…”

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

“…We note that all the regularity requirements here are indeed achievable, see, [18,19] for the details. It is possible to reduce the regularity requirement, see [20].…”

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities

“…n W x + 2lp P (D)D W l+2lp p (fl\D) (ni) For the variational inequahty it is not reahstic to assume that w G C 2 axi2~p)lv (n)C\ W 3l (f2) or w G W l + 2fp p (f2) as m the case of the équation, see [14] and [15] where conditions on the data are given for this to be achieved A more reahstic assumption for global regularity is that u G W 2 q (f2) for any q === 2, although we have not yet proved this In gênerai one can only expect u to have higher regularity either side of the f ree boundary Hence the introduction of 7" m Theorem 4 1 We now apply Theorem 4 1 to the following obstacle problem Let k satisfy Assumptions (A) with p G (1,2] K h is not a subset of K Ho wever, one can adapt the methods used in [10] to obtain similar results to the above ) In addition for u to achieve the required regularity assumptions for (4 1) and (4 2) to hold it is necessary tor the obstacle <p to satisfy these conditions, since u = <p in the contact set For example in the case of (4 2) we require that <p G C (f2) Pi W 3 l {f2) Furthermore, we would choose D= {x G f2 u(x)> <p (x)} so that F is the free boundary of the contact set Then provided u satisfies the regulanty requirements and the free boundary is regular one can apply Proof The result (4 7) with the seminorm on the left-hand side follows from (3 12) with v h = n h u and It follows from (3 9) with v h~7 r h u, …”