Abstract
An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the p-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, text {BMO} and {{,mathrm{VMO},}} spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when p=2, and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.
Highlights
We are concerned with the Dirichlet problem for the p-Laplace system− div(|∇u|p−2∇u) = − div F in (1.1) u=0 on ∂ .Here, the exponent p ∈ (1, ∞), is a bounded open set in Rn, with n 2, the function F : → RN×n, with N 1, is given, u : → RN is the unknown, and ∇u : → RN×n denotes its gradient.Under the assumption that F ∈ L p ( ), where p = p/( p − 1), one has that div F belongs to the dual of the Sobolev space W01,p( )
Our results provide an exact description of the interplay among these three pieces of information, and show that the required balance among them is qualitatively independent of the dimensions n and N, and of p
Since our results admit a local version, they provide novel optimal gradient regularity properties up to the boundary for harmonic functions vanishing on a subset of ∂. This can be regarded as a counterpart, in the scale of norms depending on oscillations, of the sharp Lq gradient regularity theory for linear equations developed in [38], and of the L∞ gradient bounds of [47]
Summary
Membership of the gradient of the solution u to problem (1.1) in Campanato type spaces depends on both the regularity of the datum F and that of the boundary ∂ in the same kind of spaces. Since our results admit a local version, they provide novel optimal gradient regularity properties up to the boundary for harmonic functions vanishing on a subset of ∂. This can be regarded as a counterpart, in the scale of norms depending on oscillations, of the sharp Lq gradient regularity theory for linear equations developed in [38], and of the L∞ gradient bounds of [47]. The continuity of u up to the boundary, even under non-homogeneous Dirichlet boundary conditions, is fully characterized thanks to the classical works [40] and [48] on a nonlinear version of the Wiener test
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