An a priori Campanato type regularity condition is established for a class of W1X local minimisers $\overline{u}$ of the general variational integral ${\int }_{\Omega }F\left(\nabla u\left(x\right)\right)\mathrm{d}x$ where $\Omega \subset {ℝ}^n$ is an open bounded domain, F is of class C2, F is strongly quasi-convex and satisfies the growth condition$F\left(\xi \right)\le c(1+|\xi {|}^p)$ for a p > 1 and where the corresponding Banach spaces X are the Morrey-Campanato space ${ℒ}^{p,\mu }(\Omega ,{ℝ}^{N\times n})$, µ < n, Campanato space ${ℒ}^{p,n}(\Omega ,{ℝ}^{N\times n})$ and the space of bounded mean oscillation $\mathrm{BMO}\Omega ,{ℝ}^{N\times n})$. The admissible maps $u:\Omega \to {ℝ}^N$ are of Sobolev class W1,p, satisfying a Dirichlet boundary...

In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.

We study the problem of minimizing ${\int }_{\Omega }F\left(Du\left(x\right)\right)dx$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\Gamma :=\partial \Omega$. The lagrangian $F$ and the domain $\Omega$ are assumed convex. A new type of hypothesis on the boundary function $\phi$ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...

We consider a function $u:\Omega \to {ℝ}^N$, $\Omega \subset {ℝ}^n$, minimizing the integral ${\int }_{\Omega }\left(\right|D_1{u|}^2+\cdots +|D_{n-1}{u|}^2+|D_{n}u{|}^p)dx$, $2(n+1)/(n+3)\le p<2$, where $D_{i}u=\partial u/\partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D\left(D_{1}u\right),\cdots ,D\left(D_{n-1}u\right)\in L^2$ and $D\left(D_{n}u\right)\in L^p$.

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is$$F\left(u\right)={\int }_{\Omega }a\left(x\right){\left[h\left(\left|Du\right|\right)\right]}^{p\left(x\right)}\mathrm{d}x$$with $h$ a convex function with general growth (also exponential behaviour is allowed).