Related papers: On the Lavrentiev gap for convex, vectorial integr…

This pre-print has now been superseded by arXiv:2305.19934 and will not be published. We prove that for convex vectorial functionals with (p,q)-growth the Lavrentiev phenomenon does not occur up to the boundary when (p,q) are suitably…

We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form: \[ \textrm{to minimize} \qquad u \mapsto \int_\Omega f(x,u,\nabla u)\,dx \,, \] under a Dirichlet boundary…

Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional, unnecessarily autonomous, defined on the space $ W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$) of Sobolev functions, possibly with prescribed one or two end point…

We establish the absence of the Lavrentiev gap between Sobolev and smooth maps for a non-autonomous variational problem of a general structure, where the integrand is assumed to be controlled by a function which is convex and anisotropic…

We consider the functional \[ F(u)=\int_{\Omega} f(\nabla u)\,dx\qquad u\in\varphi+W^{1,1}_0(\Omega) \] where $\Omega$ is a Lipschitz bounded open set of $\R^N$, $f:\R^N\to\R\cup \{+\infty\}$ is a superlinear Borel function, $\varphi\in…

Let $L:\mathbb R\times \mathbb R\to [0, +\infty[\,\cup\{+\infty\}$ be a Borel function. We consider the problem \begin{equation}\tag{P}\min F(y)=\int_0^1L(y(t), y'(t))\,dt: y(0)=0,\, y\in W^{1,1}([0,1],\mathbb R).\end{equation} We give an…

We prove that, for $1\leq p< 2$, if a $W^{1,p}$-quasiconvex integrand $\,f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ has linear growth from above on the rank-one cone, then it must satisfy this growth for all matrices in…

For a class of functionals having the $(p,q)$-growth, we establish an improved range of exponents $p$, $q$ for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution…

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form \begin{equation} \mathcal{F}(u,\Omega):=\int_\Omega \left[ f(x,Du)- g \cdot u \right] dx , \notag \end{equation}…

Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional (the "energy"), unnecessarily autonomous, defined on the space of Sobolev functions $W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$). We consider the problem of…

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,\Omega\right)=\int_{\Omega}\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,\xi)$…

We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,\Omega) = \int_{\Omega} /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon \Omega \subset {\mathbb…

We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega\to\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher…

We consider the functional $$F_\infty(u)=\int_{\Omega}f(x,u(x),\nabla u(x)) dx \quad\quad u\in \varphi+ W_0^{1,\infty}(\Omega,\mathbb{R})$$ where $\Omega$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $\varphi\in…

We prove global $W^{1,q}(\Omega,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_\Omega F(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq \psi$ for a given Sobolev obstacle $\psi$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

We prove global $W^{1,q}(\Omega,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_\Omega F(x,Du)\mathrm{d} x$. $W^{1,q}(\Omega,\mathbb{R}^m)$ regularity is also proven for minimisers of the…

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex…

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev…

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $\Omega\subset\mathbb{R}^N$ with given boundary datum $\varphi$. We prove Lipschitz regularity up to the boundary for solutions of this…