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.

On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle

In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.

Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle

<p style='text-indent:20px;'>In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.</p>

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of functionals of the form\mathbb{F}(u,\Omega)=\int_{\Omega}(F(x,Du(x))+f(x)\cdot u(x))\mathop{}\!dxwith non-autonomous integrand {F(x,\xi)} which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the partial map {x\mapsto D_{\xi}F(x,\xi)} has weak derivative in the almost critical Zygmund class {L^{n}\log^{\alpha}L} and the datum f is assumed to belong to the same Zygmund class.

A regularity result for a class of elliptic equations with lower order terms

In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

We consider functionals of the form\mathcal{F}(v,\Omega)=\int_{\Omega}f(x,Dv(x))\,dx,with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space {W^{1,q}}. We prove a higher differentiability result for the minimizers. We also infer a Lipschitz regularity result of minimizers if {q>n}, and a result of higher integrability for the gradient if {q=n}. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.