46 research outputs found
Lipschitz Regularity for a Priori Bounded Minimizers of Integral Functionals with Nonstandard Growth
Fractional differentiability for solutions of nonlinear elliptic equations
We study nonlinear elliptic equations in divergence form
When
has linear growth in , and assuming that enjoys smoothness, local
well-posedness is found in for certain values of
and . In the particular case
, and ,
, we obtain for each
. Our main tool in the proof is a more general result, that
holds also if has growth in , , and
asserts local well-posedness in for each , provided that
satisfies a locally uniform condition
Lipschitz regularity for degenerate elliptic integrals with p, q-growth
We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the x -variable
Regularity results for a class of non-differentiable obstacle problems
In this paper we prove the higher differentiability in the scale of Besov spaces of the solutions to a class of obstacle problems of the type min 2b\u3a9F(x,z,Dz):z 08K\u3c8(\u3a9). Here \u3a9 is an open bounded set of Rn, n 652, \u3c8 is a fixed function called obstacle and K\u3c8(\u3a9) is set of admissible functions z 08W1,p(\u3a9) such that z 65\u3c8 a.e. in \u3a9. We assume that the gradient of the obstacle belongs to a suitable Besov space. The main novelty here is that we are not assuming any differentiability on the partial maps x\u21a6F(x,z,Dz) and z\u21a6F(x,z,Dz), but only their H\uf6lder continuity
On very weak solutions of a class of nonlinear elliptic systems
summary:In this paper we prove a regularity result for very weak solutions of equations of the type , where , grow in the gradient like and is not in divergence form. Namely we prove that a very weak solution of our equation belongs to . We also prove global higher integrability for a very weak solution for the Dirichlet problem \cases -\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad & \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega,\Bbb R^m). \endcases $
Model problems from nonlinear elasticity: partial regularity results
In this paper we prove that every weak
and strong local
minimizer of the functional
where ,
f grows like , g grows
like and
1<q<p<2, is on an open
subset of Ω such that
. Such
functionals naturally arise from nonlinear elasticity problems. The key
point in order to obtain the partial regularity result is to
establish an energy estimate of Caccioppoli type, which is based on
an appropriate choice of the test functions. The limit case
is also treated for weak local minimizers
REGULARITY RESULTS FOR LOCAL MINIMIZERS OF FUNCTIONALS WITH DISCONTINUOUS COEFFICIENTS
We give an overview on recent regularity results of local vectorial minimizers of under two main features: the energy density is uniformly convex with respect to the gradient variable only at infinity and it depends on the spatial variable through a possibly discontinuous coefficient. More precisely, the results that we present tell that a suitable weak differentiability property of the integrand as function of the spatial variable implies the higher differentiability and the higher integrability of the gradient of the local minimizers. We also discuss the regularity of the local solutions of nonlinear elliptic equations under a fractional Sobolev assumption