Partial regularity for local minimizers of splitting-type variational integrals

We consider local minimizers u:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} of anisotropic variational integrals of (p,q)-growth with exponents 2\leq p\leq q\leq\mbox{min}\left\{ 2+p,p\frac{n}{n-2}\right\}. If the integrand is of splitting-type, then partial C^{1}-regularity of u is established

A remark on the regularity of vector-valued mappings depending on two variables which minimize splitting-type variational integrals

We combine a maximum principle for vector-valued mappings established by D’Ottavio, Leonetti and Musciano [DLM] with regularity results from [BF3] and prove the Hölder continuity of the first derivatives for local minimizers u:\Omega\rightarrow\mathbb{R}^{N} of splitting-type variational integrals provided \Omega is a domain in \mathbb{R}^{2}

Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in 2D with applications to nonlinear Hencky-materials

We prove higher integrability and differentiability results for local minimizers u:\mathbb{R}^{2}\supset\Omega\rightarrow\mathbb{R}^{M}, M\geq, of the splitting-type energy \int_{\Omega}\left[h_{1}(\left|\partial_{1}u\right|)+h_{2}(\left|\partial_{2}u\right|)\right]dx. Here h_{1}, h_{2} are rather general N-functions and no relation between hh_{1} and h_{2} is required. The methods also apply to local minimizers u:\mathbb{R}^{2}\supset\Omega\rightarrow\mathbb{R}^{2} of the functional \int_{\Omega}\left[h_{1}(\left|\textrm{div}u\right|)+h_{2}(\left|\varepsilon^{D}(u)\right|)\right]dx so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems

On the global regularity for minimizers of variational integrals : splitting-type problems in 2D and extensions to the general anisotropic setting

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain Ω ⊂ ℝ2 and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weightfunction measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefcient with respect to some improved Hölder continuity is quantifed in terms of the function dist(⋅, 휕Ω).The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli’s inequality involving small weights

Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals

We consider local minimizers u:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} of variational integrals I[u]:=\int_{\Omega}F(\nabla u)dx, where F is of anisotropic (p,q)-growth with exponents 1<p\leq q<\infty. If F is in a certain sense decomposable, we show that the dimensionless restriction q\leq2p+2 together with the local boundedness of u implies local integrability of \nabla u for all exponents t\leq p+2. More precisely, the initial exponents for the integrability of the partial derivatives can be increased by two, at least locally. If n = 2, then we use these facts to prove C^{1,\alpha}-regularity of u for any exponents 2\leq p\leq q

Variational integrals of splitting-type : higher integrability under general growth conditions

Besides other things we prove that if uin L_{loc}^{infty}(Omega;mathbb{R}^{M}),Omegasubsetmathbb{R}^{n}, locally minimizes the energy int_{Omega}left[a(left|tilde{nabla}uright|)+b(left|partial_{n}uright|)right]dx, tilde{nabla}:=(partial_{1},...,partial_{n-1}), with N-functions aleq b having the Delta_{2}-property, then left|partial_{n}uright|^{2}b(left|partial_{n}uright|)in L_{loc}^{1}(Omega). Moreover, the condition b(t)leq constt^{2}a(t^{2}) (*) for all large values of t implies left|tilde{nabla}uright|^{2}a(left|tilde{nabla}uright|)in L_{loc}^{1}(Omega). If n = 2, then these results can be improved up to left|nabla uright|in L_{loc}^{s}(Omega) for all s<infty without the hypothesis (*). If ngeq3 together with M = 1, then higher integrability for any exponent holds under more restrictive assumptions than (*)

A regularity theory for scalar local minimizers of splitting-type variational integrals

Starting from Giaquinta\u27s counterexample [Gi] we introduce the class of splitting functionals being of (p,q)-growth with exponents pleq q&#60;infty and show for the scalar case that locally bounded local minimizers are of class C^{1,mu}. Note that to our knowledge the only C^{1,mu}-results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini\u27s paper [Ma1], Theorem A, and later on in the work of Fusco and Sbordone [FS], Theorem 4.2