Boundary Regularity under Generalized Growth Conditions

Regularity under Sharp Anisotropic General Growth Conditions

We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (1.1) below, under sharp assumptions on the exponents pi in terms of p ∗: the Sobolev conjugate exponent of p; i.e., p∗ = np n−p , 1 p = 1 n Pn i=1 1 pi . As a consequence, by mean of regularity results due to Lieberman [21], we obtain the local Lipschitz-continuity of minimizers under sharp assumptio...

Gevrey Regularity for Navier–stokes Equations under Lions Boundary Conditions

The Navier–Stokes system is considered in a compact Riemannian manifold. Gevrey class regularity is proven under Lions boundary conditions in the cases of the 2D Rectangle, Cylinder, and Hemisphere. The cases of the 2D Sphere and 2D and 3D Torus are also revisited. MSC2010: 35Q30, 76D03

Boundary regularity for elliptic systems under a natural growth condition

We consider weak solutions u ∈ u0 +W 1,2 0 (Ω,R )∩L∞(Ω,RN ) of second order nonlinear elliptic systems of the type −div a( · , u,Du) = b( · , u,Du) in Ω with an inhomogeneity obeying a natural growth condition. In dimensions n ∈ {2, 3, 4} we show that Hn−1-almost every boundary point is a regular point for Du, provided that the boundary data and the coefficients are sufficiently smooth. Mathema...

Partial Regularity under Anisotropic (p,q) Growth Conditions

Critical growth biharmonic elliptic problems under Stekloff - type boundary conditions ∗

We study the fourth order nonlinear critical problem ∆u = u ∗−1 in a smooth bounded domain Ω ⊂ R, n ≥ 5, subject to the boundary conditions u = ∆u−duν = 0 on ∂Ω. We provide estimates for the range of parameters d ∈ R for which this problem admits a positive solution. If the domain is the unit ball, we obtain an almost complete description.