Related papers: Regularity for minimizers of degenerate, non-auton…

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}(\R^n)$. This criterion consists of comparing the value of a functional $\int f(u)$ with the values of the same functional applied to convolutions…

We establish sharp higher-order H\"older regularity estimates up to the boundary for solutions to equations of the form $\partial_t u-Lu=f(t,x)$ in $I\times\Omega$ where $I\subset\mathbb{R}$, $\Omega\subset\mathbb{R}^n$ and $f$ is H\"older…

In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin in 2012: \begin{align*} G_a := \inf_{u \in W_0^{1,N}(\Omega )…

We prove boundary regularity and a compactness result for parabolic nonlocal equations of the form $u_t-Iu=f$, where the operator $I$ is not necessarily translation invariant. As a consequence of this and the regularity results for…

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…

We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.

We classify local minimizers of $\int\sigma_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local…

For $n\geq 2$, $p\in(1,n)$, the "best $p$-Sobolev inequality" on an open set $\Omega\subset\mathbb{R}^n$ is identified with a family $\Phi_\Omega$ of variational problems with critical volume and trace constraints. When $\Omega$ is bounded…

We investigate the regularity of local weak solutions to evolution equations of the form \[…

A partial regularity theorem is presented for minimisers of $k$th-order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an $N$-function satisfying the $\Delta_2$…

We study families $u_s$ of functions satisfying the equations $(-\Delta)^s u_s=0$, $s \in (0,1)$ in a smooth bounded open set $\Omega \subset \mathbb{R}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of…

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $\Omega\times (0,T)$, where $\Omega\subset \mathbb{R}^N$,…

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ \text{ in }\Omega, \end{equation} where $2\leq…

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the…

In H\"ormander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the…

We consider the nonlinear problem \[(P) \;\; I u=f(x,u) \text{ in $\Omega$,} \;\; u=0 \text{ on $\mathbb{R}^{N}\setminus\Omega$ }\] in an open bounded set $\Omega\subset\mathbb{R}^{N}$, where $I$ is a nonlocal operator which may be…

The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions…

Let $(E,\|.\|)$ be a Banach space and let $(\Omega,\mu)$ be a Lebesgue measure space. We characterize, for all $p>0$, measurable functions $u:\Omega\rightarrow \mathbb{R}$ for which \begin{equation*} \left\| \int_{\Omega} f\,d\mu…

We focus on some regularity properties of $\omega$-minima of variational integrals with $\varphi$-growth and we provide an upper bound on the Hausdorff dimension of their singular set.

For higher order integral functionals with $p(x)$ growth with respect to the highest order derivative $D^m u$, we prove that $D^m u$ is H\"older continuous on an open subset $\Omega_0 \subset \Omega$ of full Lebesgue- measure, provided that…