44 research outputs found

    Fractional differentiability for solutions of nonlinear elliptic equations

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    We study nonlinear elliptic equations in divergence form divA(x,Du)=divG.{\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G. When A{\mathcal A} has linear growth in DuDu, and assuming that xA(x,ξ)x\mapsto{\mathcal A}(x,\xi) enjoys Bnα,qαB^\alpha_{\frac{n}\alpha, q} smoothness, local well-posedness is found in Bp,qαB^\alpha_{p,q} for certain values of p[2,nα)p\in[2,\frac{n}{\alpha}) and q[1,]q\in[1,\infty]. In the particular case A(x,ξ)=A(x)ξ{\mathcal A}(x,\xi)=A(x)\xi, G=0G=0 and ABnα,qαA\in B^\alpha_{\frac{n}\alpha,q}, 1q1\leq q\leq\infty, we obtain DuBp,qαDu\in B^\alpha_{p,q} for each p<nαp<\frac{n}\alpha. Our main tool in the proof is a more general result, that holds also if A{\mathcal A} has growth s1s-1 in DuDu, 2sn2\leq s\leq n, and asserts local well-posedness in LqL^q for each q>sq>s, provided that xA(x,ξ)x\mapsto{\mathcal A}(x,\xi) satisfies a locally uniform VMOVMO condition

    Lipschitz regularity for degenerate elliptic integrals with p, q-growth

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    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

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    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

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    summary:In this paper we prove a regularity result for very weak solutions of equations of the type divA(x,u,Du)=B(x,u,Du)- \operatorname{div} A(x,u,Du)=B(x, u,Du), where AA, BB grow in the gradient like tp1t^{p-1} and B(x,u,Du)B(x, u, Du) is not in divergence form. Namely we prove that a very weak solution uW1,ru\in W^{1,r} of our equation belongs to W1,pW^{1,p}. 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

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    In this paper we prove that every weak and strong local minimizer uW1,2(Ω,R3)u\in{W^{1,2}(\Omega,\mathbb{R}^3)} of the functional I(u)=ΩDu2+f(AdjDu)+g(detDu),I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du), where u:ΩR3R3 u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3, f grows like AdjDup|{\rm Adj}Du|^p, g grows like detDuq|{\rm det}Du|^q and 1<q<p<2, is C1,αC^{1,\alpha} on an open subset Ω0\Omega_0 of Ω such that meas(ΩΩ0)=0{\it meas}(\Omega\setminus \Omega_0)=0. 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 p=q2p=q\le 2 is also treated for weak local minimizers

    REGULARITY RESULTS FOR LOCAL MINIMIZERS OF FUNCTIONALS WITH DISCONTINUOUS COEFFICIENTS

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    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
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