Related papers: Regularity for higher order quasiconvex problems w…

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal…

In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence…

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs…

We show how to infer sharp partial regularity results for relaxed minimizers of degenerate, nonuniformly elliptic quasiconvex functionals, using tools from Nonlinear Potential Theory. In particular, in the setting of functionals with…

We establish partial regularity results for minimizers of a class of functionals depending on differential expressions based on elliptic operators. Specifically, we focus on functionals of Orlicz growth with a natural strong quasiconvexity…

In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$.…

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 consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected.…

We consider minimizers of linear functionals of the type $$L(u)=\int_{\p \Omega} u \, d \sigma - \int_{\Omega} u \, dx$$ in the class of convex functions $u$ with prescribed determinant $\det D^2 u =f$. We obtain compactness properties for…

We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in\phi+W^{1,1}_0(\Omega) \] where $g$ is bounded and $\phi$ satisfies the Lower Bounded…

We consider degenerate nonautonomous energies $$ \int_\Omega f(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(\Omega, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption…

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral…

We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;\Omega)= \int_{\Omega} f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty…

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem ${\rm inf}{{\rm ess sup}_{x \in \Omega} f(\nabla u(x)): u \in u_0 + W^{1,\infty}_0(\Omega)}$ when the supremand $f$ is not…

We prove partial regularity for minimizers of vectorial integrals of the Calculus of Variations, with general growth condition, imposing quasiconvexity assumptions only in an asymptotic sense.

If the smooth vector fields $X_1,\ldots,X_m$ and their commutators span the tangent space at every point in $\Omega\subseteq \mathbb{R}^N$ for any fixed $m\leq N$, then we establish the full interior regularity theory of quasi-linear…

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^n;\mathbb{R}^n)$-mappings. Then we show on the plane that this…

In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in…

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a…

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical…