Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

Abstract

We study nonlinear elliptic equations in divergence form(Formula presented.) When (Formula presented.) has linear growth in Du, and assuming that (Formula presented.) enjoys (Formula presented.) smoothness, local well-posedness is found in (Formula presented.) for certain values of (Formula presented.) and (Formula presented.). In the particular case (Formula presented.), G = 0 and (Formula presented.), (Formula presented.), we obtain (Formula presented.) for each (Formula presented.). Our main tool in the proof is a more general result, that holds also if (Formula presented.) has growth s−1 in Du, 2 ≤ s ≤ n, and asserts local well-posedness in Lq for each q > s, provided that (Formula presented.) satisfies a locally uniform VMO condition

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

“…In this context see [6], [17], [10] and recently [13], [14] and [9]. The Sobolev dependence on x recently has been considered in [22], [1] and for obstacle problems in [16].…”

“…for some positive constants M 1 , M 2 , for an exponent p ∈ (1, +∞) and for every x ∈ Ω and ξ ∈ R n . The assumption usually considered in the mathematical literature for Lipschitz continuity of solutions is the Lipschitz continuity of f (x, ξ) with respect to x; more precisely, similarly to the Dirichlet integral in (1), the condition often assumed on the x−dependence is…”

On the fractional regularity for degenerate equations with (p,q)-growth

“…Now, we give a few comments on some results which are somehow connected to Theorem 1.1. In [2], see also [6], the authors obtain fractional regularity for the solutions of a broad class of degenerate equations with nonsmooth coefficients which generalize the p-Laplacian case. Indeed, by means of a difference quotient approach and by supposing that the data relies on Besov spaces it is proved that these solutions belong to a class of Besov spaces, locally see [2, Theorems 1.1-1.3].…”

Higher differentiability for solutions of stationary p‐Stokes systems

“…On the other hand, in [33, 34] the second author observed that solutions admit a full additional derivative in the sense of if the coefficients only possess a Sobolev type regularity, see also [12, 20, 21, 23]. Finally, both types of results were unified in [3, 11], where a fractional differentiability result in the scale of Besov spaces was established under the assumption that the coefficients admit a Besov‐type regularity property. For other higher differentiability results in the scale of Besov spaces, we refer to [13].…”